Research Article Modified Biogeography-Based …downloads.hindawi.com/journals/jam/2013/960524.pdfResearch Article Modified Biogeography-Based Optimization with Local Search Mechanism

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 960524 24 pageshttpdxdoiorg1011552013960524

Research ArticleModified Biogeography-Based Optimization withLocal Search Mechanism

Quanxi Feng12 Sanyang Liu1 Qunying Wu2 GuoQiang Tang2

Haomin Zhang2 and Huazhou Chen2

1 Department of Applied Mathematics Xidian University Xirsquoan 710071 China2 School of Science Guilin University of Technology Guilin 541004 China

Correspondence should be addressed to Quanxi Feng fqx9904gmailcom and Sanyang Liu liusanyan126com

Received 3 August 2013 Revised 6 September 2013 Accepted 10 September 2013

Academic Editor Ferenc Hartung

Copyright copy 2013 Quanxi Feng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Biogeography-based optimization (BBO) is a new effective population optimization algorithm based on the biogeography theorywith inherently insufficient exploration capability To address this limitation we proposed a modified BBO with local searchmechanism (denoted as MLBBO) In MLBBO a modified migration operator is integrated into BBO which can adopt moreinformation from other habitats to enhance the exploration ability Then a local search mechanism is used in BBO to supplementwithmodifiedmigration operator Extensive experimental tests are conducted on 27 benchmark functions to show the effectivenessof the proposed algorithm The simulation results have been compared with original BBO DE improved BBO algorithms andother evolutionary algorithms Finally the performance of the modified migration operator and local search mechanism are alsodiscussed

1 Introduction

In practical application many problems are regarded as opti-mization problems Several effective techniques have beendeveloped for solving optimization problems Traditionaltechniques [1 2] are effective methods for solving theseproblems but they need to know the property of problemssuch as continuity or differentiability In the past few decadesvarious evolutionary algorithms have been sprung up forsolving complex optimization problems for example geneticalgorithm (GA) [3] evolutionary programming (EP) [4]particle swarm optimization (PSO) [5] Ant Colony opti-mization (ACO) [6] differential evolution (DE) [7] andbiogeography-based optimization (BBO) [8] Comparedwithtraditional techniques evolutionary algorithms can solveoptimization problems without using some information suchas differentiability

Biogeography-based optimization (BBO) proposed bySimon [8] is a new entrant in the domain of globaloptimization based on the theory of biogeography BBO is

developed through simulating the emigration and immi-gration of species between habitats in the multidimen-sional solution space where each habitat represents a can-didate solution Just as species in biogeography migrateback and forth between habitats features in candidatesolutions are shared between solutions through migra-tion operator Good solutions tend to share their featureswith poor solutions However these features in the ori-gin good solution may exist in several solutions bothgood and poor solutions which may weaken explorationability

Several scholars have been working for enhancing theexploration ability To mention just a few examples Gonget al [9] presented a real-coded BBO (RCBBO) for contin-uous optimization problem through the use of real code torepresent the candidate solution and integration mutationoperator to improve the population diversity Cai et al [10]proposed hybrid BBO (BBO-EP) by combing evolutionaryprogramming with BBO Boussaıd et al [11] used BBO andDE to update the population alternately and gave a two-stage

2 Journal of Applied Mathematics

(1) Begin(2) For i = 1 to NP(3) Select119867

119894in the light of immigration rate 120582

119894

(4) If 119867119894is selected

(5) For j = 1 to NP(6) Select119867

119895according to emigration rate 120583

119895

(7) If rand (01) lt 120583119895

(8) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(9) EndIf(10) EndFor(11) EndIf(12) EndFor(13) End

Algorithm 1 Pseudocode of migration operator

algorithm (DBBO) Ergezer et al [12] employed opposition-based learning alongside BBO and developed oppositionalBBO (OBBO) Ma et al [13] incorporated resampling tech-nique into BBO for solving optimization problems undernoisy environments Lohokare et al [14] proposed a newvariant of BBO called aBBOmDE In the proposed algorithma modified mutation operator and clear duplicate operatorsare used to accelerate convergence speed and the modifiedDE (mDE) is embeded as a neighborhood search operator toimprove the fitness Gong et al [15] combined the explorationofDEwith the exploitation of BBO effectively and presented ahybrid DE with BBO namely DEBBO for global numericaloptimization problem Li et al [16] designed perturbingmigration operator based on the sinusoidal migration modeland advanced perturbing biogeography-based optimization(PBBO) Inspired by multiparent crossover in evolution-ary algorithm [17] Li and Yin [18] generalized migrationoperation based on multiparent crossover and presentedmultioperator biogeography-based optimization (MOBBO)These algorithms have acquired excellent performance

It is well known that both exploration ability and exploita-tion ability are two bases for population-based optimiza-tion algorithm However the exploration and exploitationare contradictory to each other [19] How to enhance theexploration ability is a key problem for improving theperformance of BBO For the sake of this inspired of DE [7]in this paper we modify the migration operator by applyinga modified mutation strategy to enhance the explorationability Meanwhile a local search mechanism is introducedin biogeography-based algorithm to supplement with themodifiedmigration operatorWename this BBOalgorithmasmodified biogeography-based optimization with local searchmechanism (denoted asMLBBO) Analysis and experimentalresults show that MLBBO with appropriate parameters canobtain excellent performance

The rest of this paper is organized as follows In Section 2we will review the basic BBO algorithm The modifiedBBO algorithm called MLBBO is presented and analyzedin Section 3 while experimental results are presented and

discussed in Section 4 Finally conclusions and future workare drawn in Section 5

2 Review of BBO

Biogeography-based optimization (BBO) [8] is a new emerg-ing population-based algorithm proposed through mimicspeciesmigration in natural biogeography InBBOalgorithmeach possible solution is a habitat and their features thatcharacterize suitability are called suitability index variables(SIVs) [10]The goodness of each solution is named as habitatsuitability index (HSI) In BBO a habitat H is a vector ofN values (SIVs) reaching global optima through migrationstep and mutation step In original migration information isshared among habitats based on the immigration rate 120582

119894and

emigration rate 120583119894which are linear functions of the number

of species in the habitat The linear model can be calculatedas follows

120582119894= 119868 (1 minus

119894

119899)

120583119894=119864119894

119899

(1)

where E is the maximum possible emigration rate I is themaximum possible immigration rate n is the maximumnumber of species and i is the number of species of the ithsolution

The pseudocode of migration operator [8] can be looselydescribed in Algorithm 1 Where NP (= n) is the populationsize

Because of ravenous predator or some other naturalcatastrophe [20] ecological equilibrium may be destroyedwhich could lead to drastically changing the species countand HSI of habitats This phenomenon can be modeled asSIV mutation and species count probabilities are used to

Journal of Applied Mathematics 3


119894according to immigration rate 120582

119894using sinusoidal migration model


(5) For j = 1 to NP(6) Generate fourmutually different integers 1199031 1199032 1199033 and 1199034 in 1 NP(7) Select according to emigration rate 120583

119895

(8) If rand(01)lt 120583119895

(9) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(10) Else(11) 119867

119894(119878119868119881) = 119867best(119878119868119881) + 119865 sdot (1198671199031(119878119868119881) minus 1198671199032(119878119868119881)) + 119865 sdot (1198671199033(119878119868119881) minus 1198671199034(119878119868119881))


Algorithm 2 Pseudocode of modified migration operator

determine mutation rates The mutation probability mi isexpressed as follows

119898119894= 119898max sdot (

1 minus 119901119894

119901max) (2)

where 119898max is a user-defined parameter and 119901max =

argmax 119901119894 119894 = 1 2 119899

In the originalmutation operator the SIV in each solutionis probabilistically replaced with a new feature randomlygenerated in the whole solution space which will tendto increase the diversity of the population Gong et al[9] modified this mutation operator and suggested threemutation operators Gaussian mutation operator Cauchymutation operator and Levy mutation operator used in realspace

3 Modified Biogeography-Based Optimizationwith Local Search Mechanism

31 ModifiedMigration Operator Migration operator is a keyoperator of original BBO which can improve the perfor-mance of BBO Meanwhile a habitat is modified throughmigration operator by simply replacing one of the SIV of thesimilar habitat whichmeans that the new habitat absorbs lessinformation from the others Therefore migration operatorlacks exploration ability

In order to absorb more information from other habitatsinspired of DE [7] the migration operator is modified byusing a DEmutation strategy DEbesttwomutation formulais used in this paper

DEbest2 119881119894= 119883best + 119865 sdot (1198831199031 minus 1198831199032) + 119865 sdot (1198831199033 minus 1198831199034)

(3)

where 119883best is the best solution F is scaling factor119881119894119894 isin 1 2 119873119875 is the ith trial vector and

1198831199031 1198831199032 1198831199033 and 119883

1199034are for mutually different solutions

BBO has acquired excellent performance since originalmigration operator can reserve good features in the popu-lation which are avoided being loosed or destroyed in theevolution process In order to keep the exploitation ability ofBBO we modified formula (3) as follows

BBObest2 119867119894 (SIV)

= 119867best (SIV) + 119865 sdot (1198671199031 (SIV) minus 1198671199032 (SIV))

+ 119865 sdot (1198671199033 (SIV) minus 1198671199034 (SIV))

(4)

where 119867best is the best habitat Hi is the immigration habi-tat and 119867

1199031 1198671199032 1198671199033 and 119867

1199034are four mutually different

habitats From this we can see from formula (4) thattwo differences are added to the new habitat and moreinformation from other habitats are adopted which canenhance the exploration ability Meanwhile it should benoted that parameter F plays an important role in balancingthe exploration ability and exploitation ability When F takes0 formula (4) absorbs information from best habitat itsexploitation ability enhanced When F increases formula(4) absorbs more information from other habitats andits exploration ability will increase but their exploitationability will decrease a certain degree Of course this sil-ver lining has its cloud Therefore F takes 05 in thispaper

In natural biogeography the migration model may benonlinear Hence Ma [21] introduced six migration modelsComparison results show that sinusoidal migration modelis the best one The sinusoidal migration model can becalculated as follows

120582119894=119868

2(1 + cos(119894120587

119899))

120583119894=119864

2(1 minus cos(119894120587

119899))

(5)


(1) Begin(2) For i = 1 to NP(3) Compute the mutation probability p(4) Select variable119867

119894with probability p


(6) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)

(7) Endif(8) Endfor(9) End

Algorithm 3 Pseudocode of Cauchy mutation operator

In order to enhance exploration ability we modifymigration operator described in Algorithm 1 by combiningformula 119867

119894(SIV) = 119867

119895(SIV) and formula (4) The modified

migration operator is described in Algorithm 2 We can see from Algorithm 2 that immigration habitats

not only accept information from best habitat but also fromBBObest2 On the one hand information fromHSI habitatsmake the operator maintain power exploitation ability Onthe other hand information from BBObest2 can enhancethe exploration ability From this we know that this modifiedmigration operator can preserve the exploitation ability andenhance the exploration ability simultaneously

32 Mutation with Mutation Operator Cataclysmic eventscan drastically change the HSI of a natural habitat andcause species count to differ from its equilibrium value Thisphenomenon can be modeled in BBO as SIV mutation andspecies count probabilities are used to determine mutationrates

In order to enhance the exploration ability of BBO amodified mutation operator Cauchy mutation operator isintegrated into BBO The probability density function ofCauchy distribution [9] is

119891119905 (119909) =

1

120587

119905

1199052 + 1199092 (6)

where 119909 isin 119877 and 119905 gt 0 is a scale parameterTheCauchymutation (t = 1) in this paper can be described

as

119867119894 (SIV) = 119867119894 (SIV) + 120575 (1) (7)

where 120575(1) is a randomness Cauchy value when parametert equals 1 In this paper we use the Cauchy distribution toupdate the solution which can be described in Algorithm 3

33 Local Search Mechanism As Simon has pointed out theconvergence speed is an obstacle for real application Sofinding mechanisms to speed up BBO could be an importantarea for further researchTherefore a local searchmechanismis proposed to accelerate the convergence speed of BBOIn order to remedy the insufficiency of modified migration

operator which mainly updates worse habitats selected byimmigration rate the local search mechanism is designedmainly for better habitats in the first half population

For each habitat119867119894in the first half population define

119867119894 (SIV) =

119867119894 (SIV) + 119886119897119901ℎ119886 lowast (119867119896 (SIV) minus 119867119894 (SIV))

119903119886119899119889 (0 1) lt 119901119897

119867119894 (SIV) else

(8)

where 119867119896denotes a habitat randomly selected from popula-

tion 119901119897denotes the local search probability and alpha is the

strength of local search

34 Selection Operator Selection operator is used to selectbetter habitats preserving in next generationwhich can impelthemetabolismof population and retain goodhabitats In thispaper greedy selection operator is used In other words theoriginal habitat119867

119894will be replaced by its corresponding trial

habitat119872119894if the HSI of the trial habitat119872

119894is better than that

of its original habitat119867119894

119867119894= 119867119894

if 119891 (119867119894) lt 119891 (119872

119894)

119872119894

else(9)

35 Main Procedure of MLBBO Exploration ability is mainlya contributory factor to step down the performance ofBBO algorithm In order to improve the performance ofBBO a modified BBO algorithm called MLBBO is pro-posed for global optimization problems in the continu-ous domain In MLBBO the original migration opera-tor and mutation operator are modified to enhance theexploration ability Moreover local search mechanism andselection operator are integrated into MLBBO to supple-ment with migration operator The pseudo-code of MLBBOis described in Algorithm 4 where NP is the populationsize

4 Experimental Study

41 Experimental Setting In this section several experimen-tal tests are carried out on 27 benchmark functions for


(1) Begin(2) Initialize the population Pop with NP habitats randomly(3) Evaluate the fitness (HSI) for each Habitat119867

119894in Pop

(4) While the halting criteria are not satisfied do(5) Sort all the habitats from best to worst in line with their fitness values(6) Map the fitness to the number of species count S for each habitat(7) Calculate the immigration rate and emigration rate according to sinusoidal migration model(8) For i = 1 to NP lowastmigration stagelowast(9) Select119867



(10) If 119867119894is selected

(11) For j = 1 to NP(12) Generate fourmutually different integers 1199031 1199032 1199033 and 1199034 in 1 sdot sdot sdot 119873119875(13) Select119867


119895

(14) If rand(01) lt 120583119895

(15) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(16) Else(17) 119867


(18) EndIf(19) EndFor(20) EndIf(21) EndFor lowastend ofmigration stagelowast(22) For i = 1 to NP lowastmutation stagelowast(23) Compute the mutation probability 119901

119894

(24) Select variable119867119894with probability 119901

119894

(25) If 119867119894is selected

(26) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)

(27) Endif(28) EndFor lowastend ofmutation stagelowast(29) For i = 1 to NP2 lowastlocal search stagelowast(30) If rand(01) lt 119901

119897

(31) Select a habitat119867119896randomly

(32) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + alpha lowast (119867

119896(119878119868119881) minus 119867

119894(119878119868119881))

(33) EndIf(34) EndFor lowastend oflocal search stagelowast(35) Make sure each habitat legal based on boundary constraints(36) Use new generated habitat to replace duplicate(37) Evaluate the fitness (HSI) for trial habitat119872

119894in the new population Pop

(38) For i = 1 to NP lowastselection stagelowast(39) If119891(119872

119894) lt 119891(119867

119894)

(40) 119867119894= 119872119894

(41) EndIf(42) EndFor lowastend ofselection stagelowast(43) EndWhile(44) End

Algorithm 4 Pseudocode of MLBBO

validating the performance of the proposed algorithmThesefunctions consist of 13 standard benchmark functions [22](f1ndashf13) and 14 complex functions (F1ndashF14) from the set of testproblems listed in IEEE CEC 2005 [23] All the test functionsused in our experiments are high-dimensional functionswithchangeable dimension which are only briefly summarized inTable 1 A more detailed description of each function is givenin Table 8 (see also [22 23])

For fair comparison we have chosen a reasonable set ofparameter values For all experimental simulations in this

paper we will use the following parameters setting unless achange is mentioned

(1) population size NP = 100(2) maximum immigration rate I =1(3) maximum emigration rate E =1(4) mutation probability119898max = 0001(5) value to reach (VTR) = 10minus6 except for f7 F6ndashF14 of

VTR = 10minus2


Table 1 Test unctions used in our experimental tests

Function name Dimension Search Space Optimaf1 Sphere n [minus100 100] 0f2 Schwefelrsquos Problem 222 n [10 10] 0f3 Schwefelrsquos Problem 12 n [minus100 100] 0f4 Schwefelrsquos Problem 221 n [minus100 100] 0f5 Rosenbrock function n [minus30 30] 0f6 Step function n [minus100 100] 0f7 Quartic function n [minus128 128] 0f8 Schwefel function n [minus512 512] minus125695f9 Rastrigin function n [minus512 512] 0f10 Ackley function n [minus32 32] 0f11 Griewank function n [minus600 600] 0f12 Penalized function 1 n [minus50 50] 0f13 Penalized function 2 n [minus50 50] 0F1 Shifted sphere function n [minus100 100] minus450F2 Shifted Schwefelrsquos Problem 12 n [minus100 100] minus450F3 Shifted rotated high conditioned elliptic function n [minus100 100] minus450F4 Shifted Schwefelrsquos Problem 12 with noise in fitness n [minus100 100] minus450F5 Schwefelrsquos Problem 26 with global optimum on bounds n [minus30 30] minus310F6 Shifted Rosenbrockrsquos function n [minus100 100] 390F7 Shifted rotated Griewankrsquos function without bounds n Initialize in [0 600] minus180F8 Shifted rotated Ackleyrsquos function with global optimum on bounds n [minus32 32] minus140F9 Shifted Rastriginrsquos function n [minus512 512] minus330F10 Shifted Rotated Rastriginrsquos function n [minus512 512] minus330F11 Shifted Rotated Weierstrass function n [minus600 600] 90F12 Schwefelrsquos Problem 213 n [minus100 100] minus460F13 Expanded extended Griewankrsquos plus Rosenbrockrsquos function (F8F2) n [minus50 50] minus130F14 Shifted rotated expanded scafferrsquos F6 n [minus100 100] minus300

(6) maximum number of fitness function evaluations(MaxNFFEs) 150000 for f1 f6 f10 f12 and f13200000 for f2 and f11 300000 for f7 f8 f9 F1ndashF14and 500000 for f3 f4 and f5

In our experimental tests all the functions are optimizedover 30 independent runs All experimental tests are carriedout on a computer with 186GHz Intel core Processor 1 GB ofRAM and Window XP operation system in Matlab software76

42 Effect of the Strength Alpha and Rate 119901119897on the Perfor-

mance of MLBBO In this section we investigate the impactof local search strength alpha and search probability 119901

119897on

the proposed algorithm In order to analyze the performancemore scientific eight functions are chosen from Table 1Theyare Sphere Schwefel 12 Schwefel Penalized2 F1 F3 F10 and

F13 which belong to standard unimodal function standardmultimodal function shifted function shifted rotated func-tion and expanded function respectively Search strengthalpha are 03 05 08 and 11 Search rates 119901

119897are 0 02 04

06 08 and 1 It is important to note that 119901119897equal to 0

means that local search mechanism is not conducted Hencein our experiments alpha and 119901

119897are tested according to

24 situations MLBBO algorithm executes 30 independentruns on each function and convergence figures of averagefitness values (which are function error values) are plotted inFigure 1

From Figure 1 we can observe that different search rate 119901119897

have different results When 119901119897is 0 the results are worse than

the others which means that local search mechanism canimprove the performance of proposed algorithm When 119901

119897is

04 we obtain a faster convergence speed and better resultson the Sphere function For the other seven test functions


Table 2 Comparisons with BBO and DE

DE BBO MLBBO

Mean (std) MeanFEs SR Mean (Std) MeanFEs SR Mean (std) MeanFEs SR

f1 272119864 minus 20 (209119864 minus 20)+447119864 + 04 30 323119864 minus 01 (118119864 minus 01)

+ NaN 0 278119864 minus 31 (325119864 minus 31) 283119864 + 04 30


+ NaN 0 143119864 minus 21 (711119864 minus 22) 574119864 + 04 30

f3 231119864 minus 19 (237119864 minus 19)+151119864 + 05 30 126119864 + 02 (519119864 + 01)

+ NaN 0 190119864 minus 20 (303119864 minus 20) 140119864 + 05 30

f4 192119864 minus 10 (344119864 minus 10)sim261119864 + 05 30 162119864 + 00 (471119864 minus 01)

+ NaN 0 449119864 minus 08 (125119864 minus 07) 349119864 + 05 30

f5 105119864 minus 28 (892119864 minus 29)minus162119864 + 05 30 785119864 + 01 (351119864 + 01)

+ NaN 0 334119864 minus 21 (908119864 minus 21) 225119864 + 05 30

f6 867119864 minus 01 (120119864 + 00)+251119864 + 04 14 177119864 + 00 (117119864 + 00)

+119119864 + 05 3 000119864 + 00 (000119864 + 00) 114119864 + 04 30


minus221119864 + 04 30 223119864 minus 03 (991119864 minus 04) 661119864 + 04 30

f8 729119864 + 03 (232119864 + 02)+ NaN 0 253119864 minus 01 (882119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 553119864 + 04 30

f9 176119864 + 02 (101119864 + 01)+ NaN 0 356119864 minus 02 (152119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 916119864 + 04 30

f10 577119864 minus 07 (316119864 minus 06)sim787119864 + 04 29 206119864 minus 01 (494119864 minus 02)

+ NaN 0 610119864 minus 15 (114119864 minus 15) 505119864 + 04 30


+ NaN 0 287119864 minus 03 (502119864 minus 03) 333119864 + 04 22


+ NaN 0 588119864 minus 28 (322119864 minus 27) 257119864 + 04 30


+ NaN 0 509119864 minus 32 (313119864 minus 32) 271119864 + 04 30

F1 559119864 minus 28 (206119864 minus 28)+458119864 + 04 30 723119864 minus 02 (327119864 minus 02)

+ NaN 0 303119864 minus 29 (697119864 minus 29) 290119864 + 04 30

F2 759119864 minus 12 (842119864 minus 12)+172119864 + 05 30 355119864 + 03 (135119864 + 03)

+ NaN 0 197119864 minus 12 (255119864 minus 12) 158119864 + 05 30

F3 139119864 + 05 (106119864 + 05)+ NaN 0 114119864 + 07 (490119864 + 06)

+ NaN 0 926119864 + 04 (490119864 + 04) NaN 0

F4 710119864 minus 05 (161119864 minus 04)sim285119864 + 05 1 139119864 + 04 (456119864 + 03)

+ NaN 0 700119864 minus 05 (187119864 minus 04) 290119864 + 05 5

F5 531119864 + 01 (137119864 + 02)minus NaN 0 581119864 + 03 (997119864 + 02)

+ NaN 0 834119864 + 02 (381119864 + 02) NaN 0

F6 158119864 minus 13 (782119864 minus 13)sim146119864 + 05 30 223119864 + 02 (803119864 + 01)

+ NaN 0 211119864 minus 09 (668119864 minus 09) 185119864 + 05 30

F7 126119864 minus 02 (126119864 minus 02)sim493119864 + 04 15 113119864 + 00 (773119864 minus 02)

+ NaN 0 159119864 minus 02 (116119864 minus 02) 480119864 + 04 12

F8 209119864 + 01 (520119864 minus 02)sim NaN 0 208119864 + 01 (106119864 minus 01)

minus NaN 0 209119864 + 01 (707119864 minus 02) NaN 0

F9 185119864 + 02 (175119864 + 01)+ NaN 0 397119864 minus 02 (204119864 minus 02)

+ NaN 0 592119864 minus 17 (324119864 minus 16) 771119864 + 04 30

F10 205119864 + 02 (196119864 + 01)+ NaN 0 526119864 + 01 (119119864 + 01)

+ NaN 0 347119864 + 01 (124119864 + 01) NaN 0

F11 395119864 + 01 (103119864 + 00)+ NaN 0 309119864 + 01 (305119864 + 00)

+ NaN 0 172119864 + 01 (658119864 + 00) NaN 0

F12 119119864 + 03 (232119864 + 03)sim231119864 + 05 3 141119864 + 04 (984119864 + 03)

+ NaN 0 207119864 + 03 (292119864 + 03) NaN 0

F13 159119864 + 01 (112119864 + 00)+ NaN 0 109119864 + 00 (234119864 minus 01)


F14 134119864 + 01 (151119864 minus 01)+ NaN 0 131119864 + 01 (383119864 minus 01)

+ NaN 0 127119864 + 01 (240119864 minus 01) NaN 0

better 16 24

similar 9 0

worse 2 3Note ldquo+rdquo ldquominusrdquo and ldquosimrdquo indicate that MLBBO is significantly better worse and indifferent respectively compared with other algorithms

better results are obtained when search rate 119901119897is 02 and 04

Moreover the performance of MLBBO is most sensitive tosearch rate 119901

119897on function F8

From Figure 1 we can also see that search strength alphahas little effect on the results as a whole By careful looking

search strength alpha has an effect on the results when searchrate 119901

119897is large especially when 119901

119897equals 08 or 1

For considering opinions of both sides the searchstrength alpha and search rate 119901

119897are set as 08 and 02

respectively


alpha = 03alpha = 05




10minus45

10minus40

10minus35

10minus30

10minus25

10minus20

10minus15

Fitn

ess v

alue

s (lo

g)

Sphere

10minus15

10minus10

10minus5

100

105

Schwefel

Local search rate

Fitn

ess v

alue

s (lo

g)

10minus30

10minus25

10minus20

10minus15

10minus10

Schwefel2

10minus35

10minus30

10minus25

10minus20

10minus15 Penalized2

10minus28

10minus27

10minus26

10minus25

F1

104

105

106 F3

0 02 04 06 08 1Local search rate

0 02 04 06 08 1

Local search rate0 02 04 06 08 1






101321

101322

1005

1006

1007

1008

1009

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)Fi

tnes

s val

ues (

log)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

F8 F13

Figure 1 Results of MLBBO on eight test functions with different alpha and 119901119897


0 5 10 15times10

4

10minus40

10minus20

100

1020

Number of fitness function evaluations

Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1

Number of fitness function evaluationsBe

st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss

Number of fitness function evaluationsMLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

Figure 2 Convergence curves of mean fitness values of MLBBO BBO and DE for functions f1 f2 f3 f5 f6 f7 f8 f13 F1 F3 F4 F5 F7 F8F9 F11 F13 and F14


0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss

Figure 3 ANOVA tests of fitness values of MLBBO BBO and DE for functions f1 f2 f3 f5 f6 f7 f8 f13 F1 F3 F4 F5 F7 F8 F9 F11 F13and F14


Table 3 Comparisons with improved BBO algorithms

OBBO PBBO MOBBO RCBBO MLBBOMean (std) Mean (Std) Mean (std) Mean (std) Mean (std)

f1 688119864 minus 05 (258119864 minus 05)+223119864 minus 11 (277119864 minus 12)

+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

f7 402119864 minus 04 (348119864 minus 04)minus286119864 minus 04 (268119864 minus 04)

minus979119864 minus 04 (800119864 minus 04)



f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)

F1 140119864 minus 05 (826119864 minus 06)+267119864 minus 11 (969119864 minus 11)

sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)

F8 208119864 + 01 (804119864 minus 02)minus209119864 + 01 (169119864 minus 01)

minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)

Better 21 13 13 22Similar 3 10 11 2Worse 3 4 3 3Note ldquo+rdquo ldquominusrdquo and ldquosimrdquo indicate that MLBBO is significantly better worse and indifferent respectively compared with other algorithms

43 Comparisons with BBO and DEBest2Bin For evaluat-ing the performance of algorithms we first compareMLBBOwith BBO [8] which is consistent with original BBO exceptsolution representation (real code) and DEbest2bin [7](abbreviation as DE) For fair comparison three algorithmsare conducted on 27 functions with the same parameterssetting above The average and standard deviation of fitnessvalues (which are function error values) after completionof 30 Monte Carlo simulations are summarized in Table 2The number of successful runs and average of the numberof fitness function evaluations (MeanFEs) needed in eachrun when the VTR is reached within the MaxNFFEs arealso recorded in Table 2 Furthermore results of MLBBOand BBO MLBBO and DE are compared with statisticalpaired 119905-test [24 25] which is used to determine whethertwo paired solution sets differ from each other in a significantway Convergence curves of mean fitness values and boxplotfigures under 30 independent runs are listed in Figures 2 and3 respectively

FromTable 2 it is seen thatMLBBO is significantly betterthan BBO on 24 functions out of 27 test functions andMLBBO is outperformed by BBO on the rest 3 functionsWhen compared with DE MLBBO gives significantly betterperformance on 16 functions On 9 functions there are nosignificant differences between MLBBO and DE based onthe two tail 119905-test For the rest 2 functions DE are betterthan MLBBO Moreover MLBBO can obtain the VTR overall 30 successful runs within the MaxNFFEs for 16 out of27 functions Especially for the first 13 standard benchmarkfunctionsMLBBOobtains 30 successful runs on 12 functionsexcept Griewank function where 22 successful runs areobtained by MLBBO DE and BBO obtain 30 successful runson 9 and 1 out of 27 functions respectively From Table 2we can find that MLBBO needs less or similar NFFEs thanDE and BBO on most of the successful runs which meansthatMLBBO has faster convergence speed thanDE and BBOThe convergence speed and the robustness can also show inFigures 2 and 3


Table 4 Comparisons with hybrid algorithms

OXDE DEBBO MLBBOMean (std) MeanFEs SR Mean (std) MeanFEs SR Mean (std) MeanFEs SR

f1 260119864 minus 03 (904119864 minus 04)+ NaN 0 153119864 minus 30 (148119864 minus 30) + 47119864 + 04 30 230119864 minus 31 (274119864 minus 31) 28119864 + 04 30f2 182119864 minus 02 (477119864 minus 03)+ NaN 0 252119864 minus 24 (174119864 minus 24)minus 67119864 + 04 30 161119864 minus 21 (789119864 minus 22) 58119864 + 04 30f3 543119864 minus 01 (252119864 minus 01)+ NaN 0 260119864 minus 03 (172119864 minus 03)+ NaN 0 216119864 minus 20 (319119864 minus 20) 14119864 + 05 30f4 480119864 minus 02 (719119864 minus 02)+ NaN 0 650119864 minus 02 (223119864 minus 01)sim 25119864 + 05 18 799119864 minus 08 (138119864 minus 07) 35119864 + 05 30f5 373119864 minus 01 (768119864 minus 01)+ NaN 0 233119864 + 01 (910119864 + 00)+ NaN 0 256119864 minus 21 (621119864 minus 21) 23119864 + 05 30f6 000119864 + 00 (000119864 + 00)sim 93119864 + 04 30 000119864 + 00 (000119864 + 00)sim 22119864 + 04 30 000119864 + 00 (000119864 + 00) 11119864 + 04 30f7 664119864 minus 03 (197119864 minus 03)+ 20119864 + 05 30 469119864 minus 03 (154119864 minus 03)+ 14119864 + 05 30 213119864 minus 03 (910119864 minus 04) 67119864 + 04 30f8 133119864 minus 05 (157119864 minus 05)+ NaN 0 000119864 + 00 (000119864 + 00)sim 10119864 + 05 30 606119864 minus 14 (332119864 minus 13) 55119864 + 04 30f9 532119864 + 01 (672119864 + 00)+ NaN 0 000119864 + 00 (000119864 + 00)sim 18119864 + 05 30 000119864 + 00 (000119864 + 00) 92119864 + 04 30f10 145119864 minus 02 (275119864 minus 03)+ NaN 0 610119864 minus 15 (649119864 minus 16)sim 67119864 + 04 30 610119864 minus 15 (649119864 minus 16) 50119864 + 04 30f11 117119864 minus 04 (135119864 minus 04)minus NaN 0 000119864 + 00 (000119864 + 00)minus 50119864 + 04 30 337119864 minus 03 (464119864 minus 03) 28119864 + 04 19f12 697119864 minus 05 (320119864 minus 05)+ NaN 0 168119864 minus 31 (143119864 minus 31)sim 43119864 + 04 30 163119864 minus 25 (895119864 minus 25) 27119864 + 04 30f13 510119864 minus 04 (219119864 minus 04)+ NaN 0 237119864 minus 30 (269119864 minus 30)+ 47119864 + 04 30 498119864 minus 32 (419119864 minus 32) 27119864 + 04 30F1 157119864 minus 09 (748119864 minus 10)+ 24119864 + 05 30 000119864 + 00 (000119864 + 00)minus 48119864 + 04 30 269119864 minus 29 (698119864 minus 29) 29119864 + 04 30F2 608119864 + 02 (192119864 + 02)+ NaN 0 108119864 + 01 (107119864 + 01)+ NaN 0 160119864 minus 12 (157119864 minus 12) 16119864 + 05 30F3 826119864 + 06 (215119864 + 06)+ NaN 0 291119864 + 06 (101119864 + 06)+ NaN 0 900119864 + 04 (499119864 + 04) NaN 0F4 228119864 + 03 (746119864 + 02)+ NaN 0 149119864 + 02 (827119864 + 01)+ NaN 0 670119864 minus 05 (187119864 minus 04) 29119864 + 05 6F5 356119864 + 02 (942119864 + 01)minus NaN 0 117119864 + 03 (283119864 + 02)+ NaN 0 828119864 + 02 (340119864 + 02) NaN 0F6 203119864 + 01 (174119864 + 00)+ NaN 0 289119864 + 01 (161119864 + 01)+ NaN 0 176119864 minus 09 (658119864 minus 09) 19119864 + 05 30F7 352119864 minus 03 (107119864 minus 02)minus 25119864 + 05 27 764119864 minus 03 (542119864 minus 03)minus 13119864 + 05 29 147119864 minus 02 (147119864 minus 02) 50119864 + 04 19F8 209119864 + 01(572119864 minus 02)sim NaN 0 209119864 + 01 (484119864 minus 02)sim NaN 0 210119864 + 01 (454119864 minus 02) NaN 0F9 746119864 + 01 (103119864 + 01)+ NaN 0 000119864 + 00 (000119864 + 00)sim 15119864 + 05 30 000119864 + 00 (000119864 + 00) 78119864 + 04 30F10 187119864 + 02 (113119864 + 01)+ NaN 0 795119864 + 01 (919119864 + 00)+ NaN 0 337119864 + 01 (916119864 + 00) NaN 0F11 396119864 + 01 (953119864 minus 01)+ NaN 0 309119864 + 01 (135119864 + 00)+ NaN 0 183119864 + 01 (849119864 + 00) NaN 0F12 478119864 + 03 (449119864 + 03)+ NaN 0 425119864 + 03 (387119864 + 03)+ NaN 0 157119864 + 03 (187119864 + 03) NaN 0F13 108119864 + 01 (791119864 minus 01)+ NaN 0 278119864 + 00 (194119864 minus 01)minus NaN 0 305119864 + 00 (188119864 minus 01) NaN 0F14 134119864 + 01 (133119864 minus 01)+ NaN 0 129119864 + 01 (119119864 minus 01)+ NaN 0 128119864 + 01 (300119864 minus 01) NaN 0Better 22 14Similar 2 8Worse 3 5Note ldquo+rdquo ldquominusrdquo and ldquosimrdquo indicate that MLBBO is significantly better worse and indifferent respectively compared with other algorithms

From the above results we know that the total per-formance of MLBBO is significantly better than DE andBBO which indicates that MLBBO not only integrates theexploration ability of DE and the exploitation ability of BBOsimply but also can balance both search abilities very well

44 Comparisons with Improved BBOAlgorithms In this sec-tion the performance ofMLBBO is compared with improvedBBO algorithms (a) oppositional BBO (OBBO) [12] whichis proposed by employing opposition-based learning (OBL)alongside BBOrsquos migration rates (b) multi-operator BBO(MOBBO) [18] which is another modified BBO with multi-operator migration operator (c) perturb BBO (PBBO) [16]which is a modified BBO with perturb migration operatorand (d) real-code BBO (RCBBO) [9] For fair comparisonfive improved BBO algorithms are conducted on 27 functions

with the same parameters setting above The mean and stan-dard deviation of fitness values are summarized in Table 3Further the results between MLBBO and OBBO MLBBOand PBBO MLBBO andMOBBO and MLBBO and RCBBOare compared using statistical paired 119905-test Convergencecurves of mean fitness values under 30 independent runs areplotted in Figure 4

From Table 3 we can see that MLBBO outperformsOBBOon 21 out of 27 test functionswhereasOBBO surpassesMLBBO on 3 functions MLBBO obtains similar fitnessvalues with OBBO on 3 functions based on two tail 119905-testSimilar results can be obtained when making a comparisonbetween MLBBO and RCBBO When compared with PBBOandMOBBOMLBBO is significantly better than both PBBOandMOBBO on 13 functions MLBBO obtains similar resultswith PBBO andMOBBO on 10 and 11 functions respectivelyPBBO surpasses MLBBO only on 4 functions which are


Quartic function shifted rotatedAckleyrsquos function expandedextended Griewankrsquos plus Rosenbrockrsquos Function and shiftedrotated expanded Scafferrsquos F6MOBBOoutperformsMLBBOonly on 3 functions which are Quartic function shiftedrotatedAckleyrsquos function and expanded extendedGriewankrsquosplus Rosenbrockrsquos function Furthermore the total perfor-mance of MLBBO is better than PBBO and MOBBO andmuch better than OBBO and RCBBO

Furthermore Figure 4 shows similar results with Table 3We can see that the convergence speed of PBBO an MOBBOare faster than the others in preliminary stages owingto their powerful exploitation ability but the convergencespeed decreases in the next iteration However MLBBOcan converge to satisfactory solutions at equilibrium rapidlyspeed From the comparison we know that operators used inMLBBO can enhance exploration ability

45 Comparisons with Hybrid Algorithms In order to vali-date the performance of the proposed algorithms (MLBBO)further we compare the proposed algorithm with hybridalgorithmsOXDE [26] andDEBBO [15]Themean standarddeviation of fitness values of 27 test functions are listed inTable 4The averageNFFEs and successful runs are also listedin Table 4 Convergence curves of mean fitness values under30 independent runs are plotted in Figure 5

From Table 4 we can see that results of MLBBO aresignificantly better than those of OXDE in 22 out of 27high-dimensional functions while the results of OXDEoutperform those of MLBBO on only 3 functions Whencompared with DEBBO MLBBO outperforms DEBBOon 14 functions out of 27 test functions whereas DEBBOsurpasses MLBBO on 5 functions For the rest 8 functionsthere is no significant difference based on two tails 119905-testMLBBO DEBBO and OXDE obtain the VTR over all30 successful runs within the MaxNFFEs for 16 12 and 3test functions respectively From this we can come to aconclusion that MLBBO is more robust than DEBBO andOXDE which can be proved in Figure 5

Further more the average NFFEsMLBBO needed for thesuccessful run is less than DEBBO onmost functions whichcan also be indicate in Figure 5

46 Effect of Dimensionality In order to investigate theinfluence of scalability on the performance of MLBBO wecarry out a scalability study with DE original BBO DEBBO[15] and aBBOmDE [14] for scalable functions In the exper-imental study the first 13 standard benchmark functions arestudied at Dim = 10 50 100 and 200 and the other testfunctions are studied at Dim = 10 50 and 100 For Dim =10 the population size is equal to 50 and the population sizeis set to 100 for other dimensions in this paper MaxNFFEsis set to Dim lowast 10000 All the functions are optimized over30 independent runs in this paper The average and standarddeviation of fitness values of four compared algorithms aresummarized in Table 5 for standard test functions f1ndashf13 andTable 6 for CEC 2005 test functions F1ndashF14 The convergencecurves of mean fitness values under 30 independent runs arelisted in Figure 6 Furthermore the results of statistical paired

119905-test betweenMLBBO and others are also list in Tables 5 and6

It is to note that the results of DEBBO [15] for standardfunctions f1ndashf13 are taken from corresponding paper under50 independent runs The results of aBBOmDE [14] for allthe functions are taken from the paper directly which arecarried out with different population size (for Dim = 10 thepopulation size equals 30 and for other Dimensions it sets toDim) under 50 independently runs

For standard functions f1ndashf13 we can see from Table 5that the overall results are decreasing for five comparedalgorithms since increasing the problem dimensions leadsto algorithms that are sometimes unable to find the globaloptima solution within limit MaxNFFEs Similar to Dim =30 MLBBO outperforms DE on the majority of functionsand overcomes BBO on almost all the functions at everydimension MLBBO is better than or similar to DEBBO onmost of the functions

When compared with aBBOmDE we can see fromTable 5 thatMLBBO is better than or similar to aBBOmDEonmost of the functions too By carefully looking at results wecan recognize that results of some functions of MLBBO areworse than those of aBBOmDE in precision but robustnessof MLBBO is clearly better than aBBOmDE For exampleMLBBO and aBBOmDE find better solution for functionsf8 and f9 when Dim = 10 and 50 aBBOmDE fails to solvefunctions f8 and f9 when Dim increases to 100 and 200 butMLBBO can solve functions f8 and f9 as before This may bedue to modified DE mutation being used in aBBOmDE aslocal search to accelerate the convergence speed butmodifiedDE mutation is used in MLBBO as a formula to supplementwith migration operator and enhance the exploration abilityof MLBBO

ForCEC 2005 test functions F1ndashF14 we can obtain similarresults from Table 6 that MLBBO is better than or similarto DE BBO and DEBBO on most of functions whencomparing MLBBO with these algorithms However resultsof MLBBO outperform aBBOmDE on most of the functionswherever Dim = 10 or Dim = 50

From the comparison we can arrive to a conclusion thatthe total performance of MLBBO is better than the othersespecially for CEC 2005 test functions

Figure 7 shows the NFFEs required to reach VTR withdifferent dimensions for 13 standard benchmark functions(f1ndashf13) and 6 CEC 2005 test functions (F1 F2 F4 F6F7 and F9) From Figure 7 we can see that as dimensionincreases the required NFFEs increases exponentially whichmay be due to an exponentially increase in local minimawith dimension Meanwhile if we compare Figures 7(a) and7(b) we can find that CEC 2005 test functions need moreNFFEs to reach VTR than standard benchmark functionsdo which may be due to that CEC 2005 functions aremore complicated than standard benchmark functions Forexample for function f3 (Schwefel 12) F2 (Shifted Schwefel12) and F4 (Shifted Schwefel 12 with Noise in Fitness) F4F2 and f3 need 1404867 1575967 and 289960 MeanFEsto reach VTR (10minus6) respectively Obviously F4 is morecomplicated than f3 and F2 so F4 needs 289960 MeanFEsto reach VTR which is large than 1404867 and 157596


Table 5 Scalability study at Dim = 10 50 100 and 200 a ter Dim lowast 10000 NFFEs for standard functions f1ndashf13

MLBBO DE BBO DEBBO aBBOmDEMean (std) Mean (std) Mean (std) Mean (std) Mean (std)

dim = 10f1 155119864 minus 88 (653119864 minus 88) 883119864 minus 76 (189119864 minus 75)+ 214119864 minus 02 (204119864 minus 02)+ 000119864 + 00 (000119864 + 00) 149119864 minus 270 (000119864 + 00)f2 282119864 minus 46 (366119864 minus 46) 185119864 minus 36 (175119864 minus 36)+ 102119864 minus 01 (341119864 minus 02)+ 000119864 + 00 (000119864 + 00) mdashf3 637119864 minus 45 (280119864 minus 44) 854119864 minus 53 (167119864 minus 52)minus 329119864 + 00 (202119864 + 00)+ 607119864 minus 14 (960119864 minus 14) mdashf4 327119864 minus 31 (418119864 minus 31) 402119864 minus 29 (660119864 minus 29)+ 577119864 minus 01 (257119864 minus 01)minus 158119864 minus 16 (988119864 minus 17) mdashf5 105119864 minus 30 (430119864 minus 30) 000119864 + 00 (000119864 + 00)minus 143119864 + 01 (109119864 + 01)+ 340119864 + 00 (803119864 minus 01) 826119864 + 00 (169119864 + 01)f6 000119864 + 00 (000119864 + 00) 667119864 minus 02 (254119864 minus 01)minus 000119864 + 00 (000119864 + 00)sim 000119864 + 00 (000119864 + 00) mdashf7 978119864 minus 04 (739119864 minus 04) 120119864 minus 03 (819119864 minus 04)minus 652119864 minus 04 (736119864 minus 04)sim 111119864 minus 03 (396119864 minus 04) mdashf8 000119864 + 00 (000119864 + 00) 592119864 + 01 (866119864 + 01)+ 599119864 minus 02 (351119864 minus 02)+ 000119864 + 00 (000119864 + 00) 909119864 minus 14 (276119864 minus 13)f9 000119864 + 00 (000119864 + 00) 775119864 + 00 (662119864 + 00)+ 120119864 minus 02 (126119864 minus 02)+ 000119864 + 00 (000119864 + 00) 000119864 + 00 (000119864 + 00)f10 231119864 minus 15 (108119864 minus 15) 266119864 minus 15 (000119864 + 00)minus 660119864 minus 02 (258119864 minus 02)+ 589119864 minus 16 (000119864 + 00) 288119864 minus 15 (852119864 minus 16)f11 394119864 minus 03 (686119864 minus 03) 743119864 minus 02 (545119864 minus 02)+ 948119864 minus 02 (340119864 minus 02)+ 000119864 + 00 (000119864 + 00) 448119864 minus 02 (226119864 minus 02)f12 471119864 minus 32 (167119864 minus 47) 471119864 minus 32 (167119864 minus 47)minus 102119864 minus 03 (134119864 minus 03)+ 471119864 minus 32 (000119864 + 00) 471119864 minus 32 (000119864 + 00)f13 135119864 minus 32 (557119864 minus 48) 285119864 minus 32 (823119864 minus 32)minus 401119864 minus 03 (429119864 minus 03)+ 135119864 minus 32 (000119864 + 00) 135119864 minus 32 (000119864 + 00)

dim = 50f1 247119864 minus 63 (284119864 minus 63) 442119864 minus 38 (697119864 minus 38)+ 130119864 minus 01 (346119864 minus 02)+ 000119864 + 00 (000119864 + 00) 33119864 minus 154 (14119864 minus 153)f2 854119864 minus 35 (890119864 minus 35) 666119864 minus 19 (604119864 minus 19)+ 554119864 minus 01 (806119864 minus 02)+ 000119864 + 00 (000119864 + 00) mdashf3 123119864 minus 07 (149119864 minus 07) 373119864 minus 06 (623119864 minus 06)+ 787119864 + 02 (176119864 + 02)+ 520119864 minus 02 (248119864 minus 02) mdashf4 208119864 minus 02 (436119864 minus 03) 127119864 + 00 (782119864 minus 01)+ 408119864 + 00 (488119864 minus 01)+ 342119864 minus 03 (228119864 minus 02) mdashf5 225119864 minus 02 (580119864 minus 02) 106119864 + 00 (179119864 + 00)+ 144119864 + 02 (415119864 + 01)+ 443119864 + 01 (139119864 + 01) 950119864 + 00 (269119864 + 01)f6 000119864 + 00 (000119864 + 00) 587119864 + 00 (456119864 + 00)+ 100119864 minus 01 (305119864 minus 01)sim 000119864 + 00 (000119864 + 00) mdashf7 507119864 minus 03 (131119864 minus 03) 153119864 minus 02 (501119864 minus 03)+ 291119864 minus 04 (260119864 minus 04)minus 405119864 minus 03 (920119864 minus 04) mdashf8 182119864 minus 11 (000119864 + 00) 141119864 + 04 (344119864 + 02)+ 401119864 minus 01 (138119864 minus 01)+ 237119864 + 00 (167119864 + 01) 239119864 minus 11 (369119864 minus 12)f9 000119864 + 00 (000119864 + 00) 355119864 + 02 (198119864 + 01)+ 701119864 minus 02 (229119864 minus 02)+ 000119864 + 00 (000119864 + 00) 443119864 minus 14 (478119864 minus 14)f10 965119864 minus 15 (355119864 minus 15) 597119864 minus 11 (295119864 minus 10)minus 778119864 minus 02 (134119864 minus 02)+ 592119864 minus 15 (179119864 minus 15) 256119864 minus 14 (606119864 minus 15)f11 131119864 minus 03 (451119864 minus 03) 501119864 minus 03 (635119864 minus 03)+ 155119864 minus 01 (464119864 minus 02)+ 000119864 + 00 (000119864 + 00) 278119864 minus 02 (342119864 minus 02)f12 963119864 minus 33 (786119864 minus 34) 706119864 minus 02 (122119864 minus 01)+ 289119864 minus 04 (203119864 minus 04)+ 942119864 minus 33 (000119864 + 00) 942119864 minus 33 (000119864 + 00)f13 137119864 minus 32 (900119864 minus 34) 105119864 minus 15 (577119864 minus 15)minus 782119864 minus 03 (578119864 minus 03)+ 135119864 minus 32 (000119864 + 00) 135119864 minus 32 (000119864 + 00)

dim = 100f1 653119864 minus 51 (915119864 minus 51) 102119864 minus 34 (106119864 minus 34)+ 289119864 minus 01 (631119864 minus 02)+ 616119864 minus 34 (197119864 minus 33) 118119864 minus 96 (326119864 minus 96)f2 102119864 minus 25 (225119864 minus 25) 820119864 minus 19 (140119864 minus 18)+ 114119864 + 00 (118119864 minus 01)+ 000119864 + 00 (000119864 + 00) mdashf3 137119864 minus 01 (593119864 minus 02) 570119864 + 00 (193119864 + 00)+ 293119864 + 03 (486119864 + 02)+ 310119864 minus 04 (112119864 minus 04) mdashf4 452119864 minus 01 (311119864 minus 01) 299119864 + 01 (374119864 + 00)+ 601119864 + 00 (571119864 minus 01)+ 271119864 + 00 (258119864 + 00) mdashf5 389119864 + 01 (422119864 + 01) 925119864 + 01 (478119864 + 01)+ 322119864 + 02 (674119864 + 01)+ 119119864 + 02 (338119864 + 01) 257119864 + 01 (408119864 + 01)f6 000119864 + 00 (000119864 + 00) 632119864 + 01 (238119864 + 01)+ 667119864 minus 02 (254119864 minus 01)sim 000119864 + 00 (000119864 + 00) mdashf7 180119864 minus 02 (388119864 minus 03) 549119864 minus 02 (135119864 minus 02)+ 192119864 minus 04 (220119864 minus 04)- 687119864 minus 03 (115119864 minus 03) mdashf8 118119864 minus 10 (119119864 minus 11) 320119864 + 04 (523119864 + 02)+ 823119864 minus 01 (207119864 minus 01)+ 711119864 + 00 (284119864 + 01) 107119864 + 02 (113119864 + 02)f9 261119864 minus 13 (266119864 minus 13) 825119864 + 02 (472119864 + 01)+ 137119864 minus 01 (312119864 minus 02)+ 736119864 minus 01 (848119864 minus 01) 159119864 minus 01 (368119864 minus 01)f10 422119864 minus 14 (135119864 minus 14) 225119864 + 00 (492119864 minus 01)+ 826119864 minus 02 (108119864 minus 02)+ 784119864 minus 15 (703119864 minus 16) 425119864 minus 14 (799119864 minus 15)f11 540119864 minus 03 (122119864 minus 02) 369119864 minus 03 (592119864 minus 03)sim 192119864 minus 01 (499119864 minus 02)+ 000119864 + 00 (000119864 + 00) 510119864 minus 02 (889119864 minus 02)f12 189119864 minus 32 (243119864 minus 32) 580119864 minus 01 (946119864 minus 01)+ 270119864 minus 04 (270119864 minus 04)+ 471119864 minus 33 (000119864 + 00) 471119864 minus 33 (000119864 + 00)f13 337119864 minus 32 (221119864 minus 32) 115119864 + 02 (329119864 + 01)+ 108119864 minus 02 (398119864 minus 03)+ 176119864 minus 32 (268119864 minus 32) 155119864 minus 32 (598119864 minus 33)

dim = 200f1 801119864 minus 25 (123119864 minus 24) 865119864 minus 27 (272119864 minus 26)minus 798119864 minus 01 (133119864 minus 01)+ 307119864 minus 32 (248119864 minus 32) 570119864 minus 50 (734119864 minus 50)f2 512119864 minus 05 (108119864 minus 04) 358119864 minus 14 (743119864 minus 14)minus 255119864 + 00 (171119864 minus 01)+ 583119864 minus 17 (811119864 minus 17) mdashf3 639119864 + 01 (105119864 + 01) 828119864 + 02 (197119864 + 02)+ 904119864 + 03 (712119864 + 02)+ 222119864 + 05 (590119864 + 04) mdashf4 713119864 + 00 (163119864 + 00) 427119864 + 01 (311119864 + 00)+ 884119864 + 00 (649119864 minus 01)+ 159119864 + 01 (343119864 + 00) mdashf5 309119864 + 02 (635119864 + 01) 323119864 + 02 (813119864 + 01)sim 629119864 + 02 (626119864 + 01)+ 295119864 + 02 (448119864 + 01) 216119864 + 02 (998119864 + 01)f6 000119864 + 00 (000119864 + 00) 936119864 + 02 (450119864 + 02)+ 000119864 + 00 (000119864 + 00)sim 000119864 + 00 (000119864 + 00) mdash


Table 5 Continued


f7 807119864 minus 02 (124119864 minus 02) 211119864 minus 01 (540119864 minus 02)+ 146119864 minus 04 (106119864 minus 04)minus 156119864 minus 02 (282119864 minus 03) mdashf8 114119864 minus 09 (155119864 minus 09) 696119864 + 04 (612119864 + 02)+ 213119864 + 00 (330119864 minus 01)+ 201119864 + 02 (168119864 + 02) 308119864 + 02 (222119864 + 02)

f9 975119864 minus 12 (166119864 minus 11) 338119864 + 02 (206119864 + 02)+ 325119864 minus 01 (398119864 minus 02)+ 176119864 + 01 (289119864 + 00) 119119864 + 00 (752119864 minus 01)

f10 530119864 minus 13 (111119864 minus 12) 600119864 + 00 (109119864 + 00)+ 992119864 minus 02 (967119864 minus 03)+ 109119864 minus 14 (112119864 minus 15) 109119864 minus 13 (185119864 minus 14)

f11 529119864 minus 03 (173119864 minus 02) 283119864 minus 02 (757119864 minus 02)sim 288119864 minus 01 (572119864 minus 02)+ 111119864 minus 16 (000119864 + 00) 554119864 minus 02 (110119864 minus 01)

f12 190119864 minus 15 (583119864 minus 15) 236119864 + 00 (235119864 + 00)+ 309119864 minus 04 (161119864 minus 04)+ 236119864 minus 33 (000119864 + 00) 236119864 minus 33 (000119864 + 00)

f13 225119864 minus 25 (256119864 minus 25) 401119864 + 02 (511119864 + 01)+ 317119864 minus 02 (713119864 minus 03)+ 836119864 minus 32 (144119864 minus 31) 135119864 minus 32 (000119864 + 00)

Note ldquo+rdquo ldquominusrdquo and ldquosimrdquo indicate that MLBBO is significantly better worse and indifferent respectively compared with other algorithms

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss

Figure 4 Convergence curves of mean fitness values of MLBBO OBBO PBBO MOBBO and RCBBO for functions f1 f4 f8 f9 f6 f7 f8 f9f10 F1 F7 and F11


0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105

Number of fitness function evaluations0 1 2 3 4

102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss

Figure 5 Convergence curves of mean fitness values of MLBBO OXDE and DEBBO for functions f3 f5 f6 f12 F3 F5 F9 F10 and F14

47 Analysis of the Performance of Operators in MLBBO Asthe analysis above the proposed variant of BBO algorithmhas achieved excellent performance In this section weanalyze the performance of modified migration operatorand local search mechanism in MLBBO In order to ana-lyze the performance fairly and systematically we comparethe performance in four cases MLBBO with both mod-ified migration operator and local search leaning mecha-nism MLBBO with the modified migration operator andwithout local search leaning mechanism MLBBO withoutmodified migration operator and with local search leaningmechanism and MLBBO without both modified migra-tion operator and local search leaning mechanism (it isactually BBO) Four algorithms are denoted as MLBBOMLBBO2 MLBBO3 and MLBBO4 The experimental testsare conducted with 30 independent runs The parametersettings are set in Section 41 The mean standard deviation

of fitness values of experimental test are summarized inTable 7 The numbers of successful runs are also recordedin Table 7 The results between MLBBO and MLBBO2MLBBO and MLBBO3 and MLBBO and MLBBO4 arecompared using two tails 119905-test Convergence curves ofmean fitness values under 30 independent runs are listed inFigure 8

From Table 7 we can find that MLBBO is significantlybetter thanMLBBO2 on 15 functions out of 27 tests functionswhereas MLBBO2 outperforms MLBBO on 5 functions Forthe rest 7 functions there are no significant differences basedon two tails 119905-test From this we know that local searchmechanism has played an important role in improving thesearch ability especially in improving the solution precisionwhich can be proved through careful looking at fitnessvalues For example for function f1ndashf10 both algorithmsMLBBO and MLBBO2 achieve the optima but the fitness


0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne

ss(9) NFFEs (Dim = 10)

0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)

Figure 6 Mean fitness curves to compare the scalability of MLBBO DE BBO and DEBBO for selected functions (1) f2 (Dim = 100) (2)f8 (Dim = 100) (3) f12 (Dim = 100) (4) f3 (Dim = 200) (5) f10 (Dim = 200) (6) f13 (Dim = 200) (7) F1 (Dim = 10) (8) F5 (Dim = 10) (9) F8(Dim = 10) (10) F13 (Dim = 10) (11) F3 (Dim = 50) (12) F10 (Dim = 50) (13) F11 (Dim = 50) (14) F12 (Dim = 50) (15) F2 (Dim = 100) (16)F3 (Dim = 100) (17) F4 (Dim = 100) and(18) F9 (Dim = 100)


0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension

Figure 7 NFFEs versus dimensions for (a) standard functions and (b) CEC 2005 functions

0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss

Figure 8 Mean fitness curves to analyze operators of MLBBO for selected functions f1 f6 f9 F1 F3 and F11


Table 6 Scalability study at Dim = 10 50 and 100 after Dim lowast 10000 NFFEs for CEC 2005 test functions F1ndashF14

MLBBO DE BBO DEBBO aBBOmDEMean(std) Mean(std) Mean(std) Mean(std) Mean(std)

dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

F2 711119864 minus 29 (742119864 minus 29) 631119864 minus 29 (757119864 minus 29)sim373119864 + 01 (375119864 + 01)

+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)

F3 840119864 + 01 (419119864 + 02) 805119864 minus 25(916119864 minus 25)sim 237119864 + 06 (301119864 + 06)+ 186119864 + 04 (353119864 + 04)+ 102119864 + 05 (977119864 + 04)F4 728119864 minus 29 (957119864 minus 29) 391119864 minus 29 (705119864 minus 29)

sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)

F7 587119864 minus 02 (435119864 minus 02) 135119864 minus 01 (144119864 minus 01)+133119864 + 00 (404119864 minus 01)

+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)

F8 204119864 + 01 (780119864 minus 02) 203119864 + 01 (732119864 minus 02)sim204119864 + 01 (125119864 minus 01)

sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)

sim mdashF13 556119864 minus 01 (744119864 minus 02) 181119864 + 00 (608119864 minus 01)

+362119864 minus 01 (135119864 minus 01)


sim mdashF14 242119864 + 00 (492119864 minus 01) 254119864 + 00 (302119864 minus 01)

sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50

F1 825119864 minus 29 (941119864 minus 29) 177119864 minus 27 (583119864 minus 28)+118119864 minus 01 (322119864 minus 02)

+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)

F2 911119864 minus 07 (991119864 minus 07) 106119864 minus 04 (175119864 minus 04)+125119864 + 04 (369119864 + 03)

+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)

F7 141119864 minus 02 (145119864 minus 02) 845119864 minus 03 (106119864 minus 02)sim123119864 + 00 (907119864 minus 02)

+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)

F9 474119864 minus 16 (114119864 minus 15) 383119864 + 02 (357119864 + 01)+667119864 minus 02 (201119864 minus 02)

+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)

minus mdashF14 223119864 + 01 (353119864 minus 01) 230119864 + 01 (180119864 minus 01)

+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)

minus mdashF2 561119864 minus 01 (238119864 minus 01) 105119864 + 02 (454119864 + 01)

+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)

+ mdashF7 369119864 minus 03 (649119864 minus 03) 402119864 minus 03 (619119864 minus 03)

sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)

sim mdashF9 103119864 minus 14 (470119864 minus 15) 780119864 + 02 (306119864 + 02)

+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)

+ mdashNote ldquo+rdquo ldquominusrdquo and ldquosimrdquo indicate that MLBBO is significantly better worse and indifferent respectively compared with other algorithms

values obtained by MLBBO are better than those obtainedby MLBBO2 by several orders of magnitudes on thesefunctions When compared with MLBBO3 we can see thatMLBBO outperforms MLBBO3 on 19 functions out of 27test functions while MLBBO is outperformed by MLBBO3

on only 2 functions From this we know that the modifiedmigration operator has played a key role in optimizing theprocess and is better than originalmigration operator Similarconclusion can also be obtained if we compare MLBBO2 andMLBBO4 MLBBO3 and MLBBO4


Table 7 Analysis of the performance of operators in MLBBO

MLBBO MLBBO2 MLBBO3 MLBBO4Mean (std) SR Mean (std) SR Mean (std) SR Mean (std) SR

f1 203119864 minus 31 (177119864 minus 31) 30 295119864 minus 18 (105119864 minus 18)+ 30 453119864 minus 05 (265119864 minus 05)

+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0

f3 147119864 minus 20 (210119864 minus 20) 30 294119864 minus 12 (194119864 minus 12)+ 30 188119864 + 00 (610119864 minus 01)

+ 0 198119864 + 00 (563119864 minus 01)+ 0

f4 504119864 minus 08 (988119864 minus 08) 30 304119864 minus 09 (122119864 minus 09)minus 30 196119864 minus 02 (467119864 minus 03)

+ 0 232119864 minus 02 (647119864 minus 03)+ 0

f5 102119864 minus 20 (371119864 minus 20) 30 154119864 minus 14 (106119864 minus 14)+ 30 479119864 + 01 (328119864 + 01)

+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0

f11 370119864 minus 03 (602119864 minus 03) 19 000119864 + 00 (000119864 + 00)minus 30 107119864 minus 01 (687119864 minus 02)

+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0

F2 237119864 minus 12 (246119864 minus 12) 30 279119864 minus 06 (244119864 minus 06)+ 4 410119864 + 01 (107119864 + 01)

+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0

F6 193119864 minus 09 (311119864 minus 09) 30 782119864 minus 04 (405119864 minus 03)sim 29 926119864 + 02 (186119864 + 03)

+ 0 209119864 + 03 (448119864 + 03)+ 0

F7 189119864 minus 02 (173119864 minus 02) 13 156119864 minus 03 (441119864 minus 03)minus 29 167119864 minus 01 (206119864 minus 01)

+ 0 776119864 minus 01 (139119864 + 00)+ 0

F8 210119864 + 01 (538119864 minus 02) 0 209119864 + 01 (606119864 minus 02)sim 0 209119864 + 01 (410119864 minus 02)

sim 0 210119864 + 01 (380119864 minus 02)sim 0

F9 592119864 minus 17 (324119864 minus 16) 30 000119864 + 00 (000119864 + 00)sim 30 106119864 minus 03 (671119864 minus 04)

+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0

F13 308119864 + 00 (160119864 minus 01) 0 277119864 + 00 (138119864 minus 01)minus 0 104119864 + 00 (164119864 minus 01)

minus 0 982119864 minus 01 (174119864 minus 01)minus 0

F14 128119864 + 01 (238119864 minus 01) 0 130119864 + 01 (162119864 minus 01)+ 0 125119864 + 01 (265119864 minus 01)

minus 0 130119864 + 01 (186119864 minus 01)+ 0

Better 15 19 24Similar 7 6 2Worse 5 2 1Note ldquo+rdquo ldquominusrdquo and ldquosimrdquo indicate that MLBBO is significantly better worse and indifferent respectively compared with other algorithms

Furthermore if we compare four algorithms togetherthe results of MLBBO4 are worse than those of MLBBOespecially on multimodal functions which means thatthe MLBBO has more powerful exploration ability thanMLBBO4

5 Conclusions and Future Work

In this paper we have proposed a variant of modified BBOalgorithm called MLBBO for solving global optimizationproblems For origin BBO algorithm the exploration abilityis a barrier of performance of BBO We suggest a modifiedmigration operator by combining the formula in migrationoperator with a new one which can absorbmore informationfrom other habitats and then enhance the exploration abilityMoreover a local search mechanism is designed and used

in modified BBO to supplement with modified migrationoperator The proposed algorithm is compared with otherimproved BBO algorithms and evolutionary algorithmswhich show that the proposed algorithm is superior to or atleast highly competitive with the others

During the analysis we know that MLBBO possessesbetter performance and we will investigate the performanceof MLBBO in large-scale functions in the future work Wewill also investigate the built-in relationship between thepopulation diversity and exploration ability further

6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1

(lfloor119909119894+05rfloor)2

Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10

11989110(119909)=minus20exp(minus02radic1 119899

119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891

bias7119885=(119883minus119874)lowast119872

Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180

F81198658=minus20exp(minus02radic1 119899

119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13

11986513=11989111(1198915(11991111199112))+11989111(1198915(11991121199113))+sdotsdotsdot+11989111(1198915(119911119899minus1119911119899))+11989111(1198915(1199111198991199111))+119891

bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2

11986514=119865(11991111199112)+119865(11991121199113)+sdotsdotsdot+119865(119911119899minus1119911119899)+119865(1199111198991199111)+119891


Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to express thanks to the area editorand anonymous reviewers for their valuable comments andsuggestions for this paper This work is proudly supportedin part by the National Natural Science Foundation ofChina (No 11101101 No 11226219 No 61373174 and No11361019) Natural Science Foundation of Guangxi (No2013GXNSFBA019008 and No 2013GXNSFAA019003) Sci-ence and Technology Plan Project of Science and TechnologyDepartment of Hunan (No 2013FJ3083) and Project sup-ported by Program to Sponsor Teams for Innovation in theConstruction of Talent Highlands in Guangxi Institutions ofHigher Learning ([2011] 47)

References

[1] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 pp 699ndash719 1966

[2] N Andrei ldquoAccelerated scaled memoryless BFGS precondi-tioned conjugate gradient algorithm for unconstrained opti-mizationrdquo European Journal of Operational Research vol 204no 3 pp 410ndash420 2010

[3] I Ciornei and E Kyriakides ldquoHybrid ant colony-geneticalgorithm (GAAPI) for global continuous optimizationrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 1pp 234ndash245 2012

[4] G Chen C P Low and Z Yang ldquoPreserving and exploitinggenetic diversity in evolutionary programming algorithmsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 3pp 661ndash673 2009

[5] C Li S Yang and T T Nguyen ldquoA self-learning particle swarmoptimizer for global optimization problemsrdquo IEEE Transactionson Systems Man and Cybernetics B vol 42 no 3 pp 627ndash6462012

[6] S L Ho S Yang Y Bai and J Huang ldquoAn ant colony algorithmfor both robust and global optimizations of inverse problemsrdquoIEEE Transactions on Magnetics vol 49 no 5 pp 2077ndash20882013

[7] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[8] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[9] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[10] Z Cai W Gong and C X Ling ldquoResearch on a novelbiogeography-based optimization algorithm based on evolu-tionary programmingrdquo System EngineeringTheory and Practicevol 30 no 6 pp 1106ndash1112 2010

[11] I Boussaıd A Chatterjee P Siarry and M Ahmed-NacerldquoTwo-stage update biogeography-based optimization using dif-ferential evolution algorithm (DBBO)rdquo Computers and Opera-tions Research vol 38 no 8 pp 1188ndash1198 2011

[12] M Ergezer D Simon and D Du ldquoOppositional biogeography-based optimizationrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp1009ndash1014 San Antonio Tex USA October 2009

[13] H Ma M Fei D Simon and M Yu ldquoBiogeography-basedoptimization in noisy environmentsrdquo Technique Report 2012httpacademiccsuohioedusimondbbonoisy

[14] M R Lohokare S S Pattnaik B K Panigrahi and S DasldquoAccelerated biogeography-based optimization with neighbor-hood search for optimizationrdquo Applied Soft Computing vol 13no 5 pp 2318ndash2342 2013

[15] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[16] X Li J Wang J Zhou and M Yin ldquoA perturb biogeographybased optimization with mutation for global numerical opti-mizationrdquo Applied Mathematics and Computation vol 218 no2 pp 598ndash609 2011

[17] S M Elsayed R A Sarker and D L Essam ldquoMulti-operatorbased evolutionary algorithms for solving constrained opti-mization problemsrdquo Computers and Operations Research vol38 no 12 pp 1877ndash1896 2011

[18] X Li and M Yin ldquoMulti-operator based biogeography basedoptimization withmutation for global numerical optimizationrdquoComputers amp Mathematics with Applications vol 64 no 9 pp2833ndash2844 2012

[19] G Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[20] A Hastings and K Higgins ldquoPersistence of transients inspatially structured ecological modelsrdquo Science vol 263 no5150 pp 1133ndash1136 1994

[21] H Ma ldquoAn analysis of the equilibrium of migration models forbiogeography-based optimizationrdquo Information Sciences vol180 no 18 pp 3444ndash3464 2010

[22] X Yao Y Liu and G Lin ldquoEvolutionary programming madefasterrdquo IEEE Transactions on Evolutionary Computation vol 3no 2 pp 82ndash102 1999

[23] P N SuganthanNHansen J J Liang et al ldquoProblemdefinitionand evaluation criteria for the CEC 2005 special sessionon real parameter optimizationrdquo Technical Report NanyangTechnological University and KanGAL Report 2005005 IITKanpur 2005 httpwwwntuedusghomeepnsugan

[24] C H Goulden Methods of Statistical Analysis John Wiley ampSons New York NY USA 2nd edition 1956

[25] S Garcıa A Fernandez J Luengo and F Herrera ldquoAdvancednonparametric tests for multiple comparisons in the design ofexperiments in computational intelligence and data miningexperimental analysis of powerrdquo Information Sciences vol 180no 10 pp 2044ndash2064 2010

[26] Y Wang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of



119894in the light of immigration rate 120582

119894


(5) For j = 1 to NP(6) Select119867


119895

(7) If rand (01) lt 120583119895

(8) 119867119894(119878119868119881) = 119867

119895(119878119868119881)


Algorithm 1 Pseudocode of migration operator

algorithm (DBBO) Ergezer et al [12] employed opposition-based learning alongside BBO and developed oppositionalBBO (OBBO) Ma et al [13] incorporated resampling tech-nique into BBO for solving optimization problems undernoisy environments Lohokare et al [14] proposed a newvariant of BBO called aBBOmDE In the proposed algorithma modified mutation operator and clear duplicate operatorsare used to accelerate convergence speed and the modifiedDE (mDE) is embeded as a neighborhood search operator toimprove the fitness Gong et al [15] combined the explorationofDEwith the exploitation of BBO effectively and presented ahybrid DE with BBO namely DEBBO for global numericaloptimization problem Li et al [16] designed perturbingmigration operator based on the sinusoidal migration modeland advanced perturbing biogeography-based optimization(PBBO) Inspired by multiparent crossover in evolution-ary algorithm [17] Li and Yin [18] generalized migrationoperation based on multiparent crossover and presentedmultioperator biogeography-based optimization (MOBBO)These algorithms have acquired excellent performance

It is well known that both exploration ability and exploita-tion ability are two bases for population-based optimiza-tion algorithm However the exploration and exploitationare contradictory to each other [19] How to enhance theexploration ability is a key problem for improving theperformance of BBO For the sake of this inspired of DE [7]in this paper we modify the migration operator by applyinga modified mutation strategy to enhance the explorationability Meanwhile a local search mechanism is introducedin biogeography-based algorithm to supplement with themodifiedmigration operatorWename this BBOalgorithmasmodified biogeography-based optimization with local searchmechanism (denoted asMLBBO) Analysis and experimentalresults show that MLBBO with appropriate parameters canobtain excellent performance

The rest of this paper is organized as follows In Section 2we will review the basic BBO algorithm The modifiedBBO algorithm called MLBBO is presented and analyzedin Section 3 while experimental results are presented and

discussed in Section 4 Finally conclusions and future workare drawn in Section 5

2 Review of BBO

Biogeography-based optimization (BBO) [8] is a new emerg-ing population-based algorithm proposed through mimicspeciesmigration in natural biogeography InBBOalgorithmeach possible solution is a habitat and their features thatcharacterize suitability are called suitability index variables(SIVs) [10]The goodness of each solution is named as habitatsuitability index (HSI) In BBO a habitat H is a vector ofN values (SIVs) reaching global optima through migrationstep and mutation step In original migration information isshared among habitats based on the immigration rate 120582

119894and

emigration rate 120583119894which are linear functions of the number

of species in the habitat The linear model can be calculatedas follows

120582119894= 119868 (1 minus

119894

119899)

120583119894=119864119894

119899

(1)

where E is the maximum possible emigration rate I is themaximum possible immigration rate n is the maximumnumber of species and i is the number of species of the ithsolution

The pseudocode of migration operator [8] can be looselydescribed in Algorithm 1 Where NP (= n) is the populationsize

Because of ravenous predator or some other naturalcatastrophe [20] ecological equilibrium may be destroyedwhich could lead to drastically changing the species countand HSI of habitats This phenomenon can be modeled asSIV mutation and species count probabilities are used to







119895

(8) If rand(01)lt 120583119895

(9) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(10) Else(11) 119867





119898119894= 119898max sdot (

1 minus 119901119894

119901max) (2)


argmax 119901119894 119894 = 1 2 119899






(3)


1198831199031 1198831199032 1198831199033 and 119883



BBObest2 119867119894 (SIV)



(4)


1199031 1198671199032 1198671199033 and 119867




120582119894=119868

2(1 + cos(119894120587

119899))

120583119894=119864

2(1 minus cos(119894120587

119899))

(5)





(6) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)




119894(SIV) = 119867






119891119905 (119909) =

1

120587

119905

1199052 + 1199092 (6)


as

119867119894 (SIV) = 119867119894 (SIV) + 120575 (1) (7)





119867119894 (SIV) =


119903119886119899119889 (0 1) lt 119901119897

119867119894 (SIV) else

(8)









119867119894= 119867119894

if 119891 (119867119894) lt 119891 (119872

119894)

119872119894

else(9)






119894in Pop




(10) If 119867119894is selected



119895

(14) If rand(01) lt 120583119895

(15) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(16) Else(17) 119867



119894


119894

(25) If 119867119894is selected

(26) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)


119897


(32) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + alpha lowast (119867

119896(119878119868119881) minus 119867

119894(119878119868119881))




119894) lt 119891(119867

119894)

(40) 119867119894= 119872119894







VTR = 10minus2








119897on



119897are 0 02 04








119897is




DE BBO MLBBO



+ NaN 0 278119864 minus 31 (325119864 minus 31) 283119864 + 04 30


+ NaN 0 143119864 minus 21 (711119864 minus 22) 574119864 + 04 30

f3 231119864 minus 19 (237119864 minus 19)+151119864 + 05 30 126119864 + 02 (519119864 + 01)

+ NaN 0 190119864 minus 20 (303119864 minus 20) 140119864 + 05 30


+ NaN 0 449119864 minus 08 (125119864 minus 07) 349119864 + 05 30


+ NaN 0 334119864 minus 21 (908119864 minus 21) 225119864 + 05 30

f6 867119864 minus 01 (120119864 + 00)+251119864 + 04 14 177119864 + 00 (117119864 + 00)

+119119864 + 05 3 000119864 + 00 (000119864 + 00) 114119864 + 04 30



f8 729119864 + 03 (232119864 + 02)+ NaN 0 253119864 minus 01 (882119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 553119864 + 04 30

f9 176119864 + 02 (101119864 + 01)+ NaN 0 356119864 minus 02 (152119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 916119864 + 04 30


+ NaN 0 610119864 minus 15 (114119864 minus 15) 505119864 + 04 30


+ NaN 0 287119864 minus 03 (502119864 minus 03) 333119864 + 04 22


+ NaN 0 588119864 minus 28 (322119864 minus 27) 257119864 + 04 30


+ NaN 0 509119864 minus 32 (313119864 minus 32) 271119864 + 04 30


+ NaN 0 303119864 minus 29 (697119864 minus 29) 290119864 + 04 30

F2 759119864 minus 12 (842119864 minus 12)+172119864 + 05 30 355119864 + 03 (135119864 + 03)

+ NaN 0 197119864 minus 12 (255119864 minus 12) 158119864 + 05 30

F3 139119864 + 05 (106119864 + 05)+ NaN 0 114119864 + 07 (490119864 + 06)

+ NaN 0 926119864 + 04 (490119864 + 04) NaN 0

F4 710119864 minus 05 (161119864 minus 04)sim285119864 + 05 1 139119864 + 04 (456119864 + 03)

+ NaN 0 700119864 minus 05 (187119864 minus 04) 290119864 + 05 5

F5 531119864 + 01 (137119864 + 02)minus NaN 0 581119864 + 03 (997119864 + 02)

+ NaN 0 834119864 + 02 (381119864 + 02) NaN 0

F6 158119864 minus 13 (782119864 minus 13)sim146119864 + 05 30 223119864 + 02 (803119864 + 01)

+ NaN 0 211119864 minus 09 (668119864 minus 09) 185119864 + 05 30


+ NaN 0 159119864 minus 02 (116119864 minus 02) 480119864 + 04 12



F9 185119864 + 02 (175119864 + 01)+ NaN 0 397119864 minus 02 (204119864 minus 02)

+ NaN 0 592119864 minus 17 (324119864 minus 16) 771119864 + 04 30

F10 205119864 + 02 (196119864 + 01)+ NaN 0 526119864 + 01 (119119864 + 01)

+ NaN 0 347119864 + 01 (124119864 + 01) NaN 0

F11 395119864 + 01 (103119864 + 00)+ NaN 0 309119864 + 01 (305119864 + 00)

+ NaN 0 172119864 + 01 (658119864 + 00) NaN 0

F12 119119864 + 03 (232119864 + 03)sim231119864 + 05 3 141119864 + 04 (984119864 + 03)

+ NaN 0 207119864 + 03 (292119864 + 03) NaN 0

F13 159119864 + 01 (112119864 + 00)+ NaN 0 109119864 + 00 (234119864 minus 01)


F14 134119864 + 01 (151119864 minus 01)+ NaN 0 131119864 + 01 (383119864 minus 01)

+ NaN 0 127119864 + 01 (240119864 minus 01) NaN 0

better 16 24

similar 9 0











respectively






10minus45

10minus40

10minus35

10minus30

10minus25

10minus20

10minus15

Fitn

ess v

alue

s (lo

g)

Sphere

10minus15

10minus10

10minus5

100

105

Schwefel

Local search rate

Fitn

ess v

alue

s (lo

g)

10minus30

10minus25

10minus20

10minus15

10minus10

Schwefel2

10minus35

10minus30

10minus25

10minus20


10minus28

10minus27

10minus26

10minus25

F1

104

105

106 F3


0 02 04 06 08 1







101321

101322

1005

1006

1007

1008

1009

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)Fi

tnes

s val

ues (

log)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

F8 F13



0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1


st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO



0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119895

(8) If rand(01)lt 120583119895

(9) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(10) Else(11) 119867





119898119894= 119898max sdot (

1 minus 119901119894

119901max) (2)


argmax 119901119894 119894 = 1 2 119899






(3)


1198831199031 1198831199032 1198831199033 and 119883



BBObest2 119867119894 (SIV)



(4)


1199031 1198671199032 1198671199033 and 119867




120582119894=119868

2(1 + cos(119894120587

119899))

120583119894=119864

2(1 minus cos(119894120587

119899))

(5)





(6) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)




119894(SIV) = 119867






119891119905 (119909) =

1

120587

119905

1199052 + 1199092 (6)


as

119867119894 (SIV) = 119867119894 (SIV) + 120575 (1) (7)





119867119894 (SIV) =


119903119886119899119889 (0 1) lt 119901119897

119867119894 (SIV) else

(8)









119867119894= 119867119894

if 119891 (119867119894) lt 119891 (119872

119894)

119872119894

else(9)






119894in Pop




(10) If 119867119894is selected



119895

(14) If rand(01) lt 120583119895

(15) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(16) Else(17) 119867



119894


119894

(25) If 119867119894is selected

(26) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)


119897


(32) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + alpha lowast (119867

119896(119878119868119881) minus 119867

119894(119878119868119881))




119894) lt 119891(119867

119894)

(40) 119867119894= 119872119894







VTR = 10minus2








119897on



119897are 0 02 04








119897is




DE BBO MLBBO



+ NaN 0 278119864 minus 31 (325119864 minus 31) 283119864 + 04 30


+ NaN 0 143119864 minus 21 (711119864 minus 22) 574119864 + 04 30

f3 231119864 minus 19 (237119864 minus 19)+151119864 + 05 30 126119864 + 02 (519119864 + 01)

+ NaN 0 190119864 minus 20 (303119864 minus 20) 140119864 + 05 30


+ NaN 0 449119864 minus 08 (125119864 minus 07) 349119864 + 05 30


+ NaN 0 334119864 minus 21 (908119864 minus 21) 225119864 + 05 30

f6 867119864 minus 01 (120119864 + 00)+251119864 + 04 14 177119864 + 00 (117119864 + 00)

+119119864 + 05 3 000119864 + 00 (000119864 + 00) 114119864 + 04 30



f8 729119864 + 03 (232119864 + 02)+ NaN 0 253119864 minus 01 (882119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 553119864 + 04 30

f9 176119864 + 02 (101119864 + 01)+ NaN 0 356119864 minus 02 (152119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 916119864 + 04 30


+ NaN 0 610119864 minus 15 (114119864 minus 15) 505119864 + 04 30


+ NaN 0 287119864 minus 03 (502119864 minus 03) 333119864 + 04 22


+ NaN 0 588119864 minus 28 (322119864 minus 27) 257119864 + 04 30


+ NaN 0 509119864 minus 32 (313119864 minus 32) 271119864 + 04 30


+ NaN 0 303119864 minus 29 (697119864 minus 29) 290119864 + 04 30

F2 759119864 minus 12 (842119864 minus 12)+172119864 + 05 30 355119864 + 03 (135119864 + 03)

+ NaN 0 197119864 minus 12 (255119864 minus 12) 158119864 + 05 30

F3 139119864 + 05 (106119864 + 05)+ NaN 0 114119864 + 07 (490119864 + 06)

+ NaN 0 926119864 + 04 (490119864 + 04) NaN 0

F4 710119864 minus 05 (161119864 minus 04)sim285119864 + 05 1 139119864 + 04 (456119864 + 03)

+ NaN 0 700119864 minus 05 (187119864 minus 04) 290119864 + 05 5

F5 531119864 + 01 (137119864 + 02)minus NaN 0 581119864 + 03 (997119864 + 02)

+ NaN 0 834119864 + 02 (381119864 + 02) NaN 0

F6 158119864 minus 13 (782119864 minus 13)sim146119864 + 05 30 223119864 + 02 (803119864 + 01)

+ NaN 0 211119864 minus 09 (668119864 minus 09) 185119864 + 05 30


+ NaN 0 159119864 minus 02 (116119864 minus 02) 480119864 + 04 12



F9 185119864 + 02 (175119864 + 01)+ NaN 0 397119864 minus 02 (204119864 minus 02)

+ NaN 0 592119864 minus 17 (324119864 minus 16) 771119864 + 04 30

F10 205119864 + 02 (196119864 + 01)+ NaN 0 526119864 + 01 (119119864 + 01)

+ NaN 0 347119864 + 01 (124119864 + 01) NaN 0

F11 395119864 + 01 (103119864 + 00)+ NaN 0 309119864 + 01 (305119864 + 00)

+ NaN 0 172119864 + 01 (658119864 + 00) NaN 0

F12 119119864 + 03 (232119864 + 03)sim231119864 + 05 3 141119864 + 04 (984119864 + 03)

+ NaN 0 207119864 + 03 (292119864 + 03) NaN 0

F13 159119864 + 01 (112119864 + 00)+ NaN 0 109119864 + 00 (234119864 minus 01)


F14 134119864 + 01 (151119864 minus 01)+ NaN 0 131119864 + 01 (383119864 minus 01)

+ NaN 0 127119864 + 01 (240119864 minus 01) NaN 0

better 16 24

similar 9 0











respectively






10minus45

10minus40

10minus35

10minus30

10minus25

10minus20

10minus15

Fitn

ess v

alue

s (lo

g)

Sphere

10minus15

10minus10

10minus5

100

105

Schwefel

Local search rate

Fitn

ess v

alue

s (lo

g)

10minus30

10minus25

10minus20

10minus15

10minus10

Schwefel2

10minus35

10minus30

10minus25

10minus20


10minus28

10minus27

10minus26

10minus25

F1

104

105

106 F3


0 02 04 06 08 1







101321

101322

1005

1006

1007

1008

1009

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)Fi

tnes

s val

ues (

log)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

F8 F13



0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1


st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO



0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(6) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)




119894(SIV) = 119867






119891119905 (119909) =

1

120587

119905

1199052 + 1199092 (6)


as

119867119894 (SIV) = 119867119894 (SIV) + 120575 (1) (7)





119867119894 (SIV) =


119903119886119899119889 (0 1) lt 119901119897

119867119894 (SIV) else

(8)









119867119894= 119867119894

if 119891 (119867119894) lt 119891 (119872

119894)

119872119894

else(9)






119894in Pop




(10) If 119867119894is selected



119895

(14) If rand(01) lt 120583119895

(15) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(16) Else(17) 119867



119894


119894

(25) If 119867119894is selected

(26) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)


119897


(32) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + alpha lowast (119867

119896(119878119868119881) minus 119867

119894(119878119868119881))




119894) lt 119891(119867

119894)

(40) 119867119894= 119872119894







VTR = 10minus2








119897on



119897are 0 02 04








119897is




DE BBO MLBBO



+ NaN 0 278119864 minus 31 (325119864 minus 31) 283119864 + 04 30


+ NaN 0 143119864 minus 21 (711119864 minus 22) 574119864 + 04 30

f3 231119864 minus 19 (237119864 minus 19)+151119864 + 05 30 126119864 + 02 (519119864 + 01)

+ NaN 0 190119864 minus 20 (303119864 minus 20) 140119864 + 05 30


+ NaN 0 449119864 minus 08 (125119864 minus 07) 349119864 + 05 30


+ NaN 0 334119864 minus 21 (908119864 minus 21) 225119864 + 05 30

f6 867119864 minus 01 (120119864 + 00)+251119864 + 04 14 177119864 + 00 (117119864 + 00)

+119119864 + 05 3 000119864 + 00 (000119864 + 00) 114119864 + 04 30



f8 729119864 + 03 (232119864 + 02)+ NaN 0 253119864 minus 01 (882119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 553119864 + 04 30

f9 176119864 + 02 (101119864 + 01)+ NaN 0 356119864 minus 02 (152119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 916119864 + 04 30


+ NaN 0 610119864 minus 15 (114119864 minus 15) 505119864 + 04 30


+ NaN 0 287119864 minus 03 (502119864 minus 03) 333119864 + 04 22


+ NaN 0 588119864 minus 28 (322119864 minus 27) 257119864 + 04 30


+ NaN 0 509119864 minus 32 (313119864 minus 32) 271119864 + 04 30


+ NaN 0 303119864 minus 29 (697119864 minus 29) 290119864 + 04 30

F2 759119864 minus 12 (842119864 minus 12)+172119864 + 05 30 355119864 + 03 (135119864 + 03)

+ NaN 0 197119864 minus 12 (255119864 minus 12) 158119864 + 05 30

F3 139119864 + 05 (106119864 + 05)+ NaN 0 114119864 + 07 (490119864 + 06)

+ NaN 0 926119864 + 04 (490119864 + 04) NaN 0

F4 710119864 minus 05 (161119864 minus 04)sim285119864 + 05 1 139119864 + 04 (456119864 + 03)

+ NaN 0 700119864 minus 05 (187119864 minus 04) 290119864 + 05 5

F5 531119864 + 01 (137119864 + 02)minus NaN 0 581119864 + 03 (997119864 + 02)

+ NaN 0 834119864 + 02 (381119864 + 02) NaN 0

F6 158119864 minus 13 (782119864 minus 13)sim146119864 + 05 30 223119864 + 02 (803119864 + 01)

+ NaN 0 211119864 minus 09 (668119864 minus 09) 185119864 + 05 30


+ NaN 0 159119864 minus 02 (116119864 minus 02) 480119864 + 04 12



F9 185119864 + 02 (175119864 + 01)+ NaN 0 397119864 minus 02 (204119864 minus 02)

+ NaN 0 592119864 minus 17 (324119864 minus 16) 771119864 + 04 30

F10 205119864 + 02 (196119864 + 01)+ NaN 0 526119864 + 01 (119119864 + 01)

+ NaN 0 347119864 + 01 (124119864 + 01) NaN 0

F11 395119864 + 01 (103119864 + 00)+ NaN 0 309119864 + 01 (305119864 + 00)

+ NaN 0 172119864 + 01 (658119864 + 00) NaN 0

F12 119119864 + 03 (232119864 + 03)sim231119864 + 05 3 141119864 + 04 (984119864 + 03)

+ NaN 0 207119864 + 03 (292119864 + 03) NaN 0

F13 159119864 + 01 (112119864 + 00)+ NaN 0 109119864 + 00 (234119864 minus 01)


F14 134119864 + 01 (151119864 minus 01)+ NaN 0 131119864 + 01 (383119864 minus 01)

+ NaN 0 127119864 + 01 (240119864 minus 01) NaN 0

better 16 24

similar 9 0











respectively






10minus45

10minus40

10minus35

10minus30

10minus25

10minus20

10minus15

Fitn

ess v

alue

s (lo

g)

Sphere

10minus15

10minus10

10minus5

100

105

Schwefel

Local search rate

Fitn

ess v

alue

s (lo

g)

10minus30

10minus25

10minus20

10minus15

10minus10

Schwefel2

10minus35

10minus30

10minus25

10minus20


10minus28

10minus27

10minus26

10minus25

F1

104

105

106 F3


0 02 04 06 08 1







101321

101322

1005

1006

1007

1008

1009

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)Fi

tnes

s val

ues (

log)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

F8 F13



0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1


st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO



0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894in Pop




(10) If 119867119894is selected



119895

(14) If rand(01) lt 120583119895

(15) 119867119894(119878119868119881) = 119867

119895(119878119868119881)

(16) Else(17) 119867



119894


119894

(25) If 119867119894is selected

(26) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + 120575(1)


119897


(32) 119867119894(119878119868119881) = 119867

119894(119878119868119881) + alpha lowast (119867

119896(119878119868119881) minus 119867

119894(119878119868119881))




119894) lt 119891(119867

119894)

(40) 119867119894= 119872119894







VTR = 10minus2








119897on



119897are 0 02 04








119897is




DE BBO MLBBO



+ NaN 0 278119864 minus 31 (325119864 minus 31) 283119864 + 04 30


+ NaN 0 143119864 minus 21 (711119864 minus 22) 574119864 + 04 30

f3 231119864 minus 19 (237119864 minus 19)+151119864 + 05 30 126119864 + 02 (519119864 + 01)

+ NaN 0 190119864 minus 20 (303119864 minus 20) 140119864 + 05 30


+ NaN 0 449119864 minus 08 (125119864 minus 07) 349119864 + 05 30


+ NaN 0 334119864 minus 21 (908119864 minus 21) 225119864 + 05 30

f6 867119864 minus 01 (120119864 + 00)+251119864 + 04 14 177119864 + 00 (117119864 + 00)

+119119864 + 05 3 000119864 + 00 (000119864 + 00) 114119864 + 04 30



f8 729119864 + 03 (232119864 + 02)+ NaN 0 253119864 minus 01 (882119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 553119864 + 04 30

f9 176119864 + 02 (101119864 + 01)+ NaN 0 356119864 minus 02 (152119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 916119864 + 04 30


+ NaN 0 610119864 minus 15 (114119864 minus 15) 505119864 + 04 30


+ NaN 0 287119864 minus 03 (502119864 minus 03) 333119864 + 04 22


+ NaN 0 588119864 minus 28 (322119864 minus 27) 257119864 + 04 30


+ NaN 0 509119864 minus 32 (313119864 minus 32) 271119864 + 04 30


+ NaN 0 303119864 minus 29 (697119864 minus 29) 290119864 + 04 30

F2 759119864 minus 12 (842119864 minus 12)+172119864 + 05 30 355119864 + 03 (135119864 + 03)

+ NaN 0 197119864 minus 12 (255119864 minus 12) 158119864 + 05 30

F3 139119864 + 05 (106119864 + 05)+ NaN 0 114119864 + 07 (490119864 + 06)

+ NaN 0 926119864 + 04 (490119864 + 04) NaN 0

F4 710119864 minus 05 (161119864 minus 04)sim285119864 + 05 1 139119864 + 04 (456119864 + 03)

+ NaN 0 700119864 minus 05 (187119864 minus 04) 290119864 + 05 5

F5 531119864 + 01 (137119864 + 02)minus NaN 0 581119864 + 03 (997119864 + 02)

+ NaN 0 834119864 + 02 (381119864 + 02) NaN 0

F6 158119864 minus 13 (782119864 minus 13)sim146119864 + 05 30 223119864 + 02 (803119864 + 01)

+ NaN 0 211119864 minus 09 (668119864 minus 09) 185119864 + 05 30


+ NaN 0 159119864 minus 02 (116119864 minus 02) 480119864 + 04 12



F9 185119864 + 02 (175119864 + 01)+ NaN 0 397119864 minus 02 (204119864 minus 02)

+ NaN 0 592119864 minus 17 (324119864 minus 16) 771119864 + 04 30

F10 205119864 + 02 (196119864 + 01)+ NaN 0 526119864 + 01 (119119864 + 01)

+ NaN 0 347119864 + 01 (124119864 + 01) NaN 0

F11 395119864 + 01 (103119864 + 00)+ NaN 0 309119864 + 01 (305119864 + 00)

+ NaN 0 172119864 + 01 (658119864 + 00) NaN 0

F12 119119864 + 03 (232119864 + 03)sim231119864 + 05 3 141119864 + 04 (984119864 + 03)

+ NaN 0 207119864 + 03 (292119864 + 03) NaN 0

F13 159119864 + 01 (112119864 + 00)+ NaN 0 109119864 + 00 (234119864 minus 01)


F14 134119864 + 01 (151119864 minus 01)+ NaN 0 131119864 + 01 (383119864 minus 01)

+ NaN 0 127119864 + 01 (240119864 minus 01) NaN 0

better 16 24

similar 9 0











respectively






10minus45

10minus40

10minus35

10minus30

10minus25

10minus20

10minus15

Fitn

ess v

alue

s (lo

g)

Sphere

10minus15

10minus10

10minus5

100

105

Schwefel

Local search rate

Fitn

ess v

alue

s (lo

g)

10minus30

10minus25

10minus20

10minus15

10minus10

Schwefel2

10minus35

10minus30

10minus25

10minus20


10minus28

10minus27

10minus26

10minus25

F1

104

105

106 F3


0 02 04 06 08 1







101321

101322

1005

1006

1007

1008

1009

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)Fi

tnes

s val

ues (

log)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

F8 F13



0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1


st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO



0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119897on



119897are 0 02 04








119897is




DE BBO MLBBO



+ NaN 0 278119864 minus 31 (325119864 minus 31) 283119864 + 04 30


+ NaN 0 143119864 minus 21 (711119864 minus 22) 574119864 + 04 30

f3 231119864 minus 19 (237119864 minus 19)+151119864 + 05 30 126119864 + 02 (519119864 + 01)

+ NaN 0 190119864 minus 20 (303119864 minus 20) 140119864 + 05 30


+ NaN 0 449119864 minus 08 (125119864 minus 07) 349119864 + 05 30


+ NaN 0 334119864 minus 21 (908119864 minus 21) 225119864 + 05 30

f6 867119864 minus 01 (120119864 + 00)+251119864 + 04 14 177119864 + 00 (117119864 + 00)

+119119864 + 05 3 000119864 + 00 (000119864 + 00) 114119864 + 04 30



f8 729119864 + 03 (232119864 + 02)+ NaN 0 253119864 minus 01 (882119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 553119864 + 04 30

f9 176119864 + 02 (101119864 + 01)+ NaN 0 356119864 minus 02 (152119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 916119864 + 04 30


+ NaN 0 610119864 minus 15 (114119864 minus 15) 505119864 + 04 30


+ NaN 0 287119864 minus 03 (502119864 minus 03) 333119864 + 04 22


+ NaN 0 588119864 minus 28 (322119864 minus 27) 257119864 + 04 30


+ NaN 0 509119864 minus 32 (313119864 minus 32) 271119864 + 04 30


+ NaN 0 303119864 minus 29 (697119864 minus 29) 290119864 + 04 30

F2 759119864 minus 12 (842119864 minus 12)+172119864 + 05 30 355119864 + 03 (135119864 + 03)

+ NaN 0 197119864 minus 12 (255119864 minus 12) 158119864 + 05 30

F3 139119864 + 05 (106119864 + 05)+ NaN 0 114119864 + 07 (490119864 + 06)

+ NaN 0 926119864 + 04 (490119864 + 04) NaN 0

F4 710119864 minus 05 (161119864 minus 04)sim285119864 + 05 1 139119864 + 04 (456119864 + 03)

+ NaN 0 700119864 minus 05 (187119864 minus 04) 290119864 + 05 5

F5 531119864 + 01 (137119864 + 02)minus NaN 0 581119864 + 03 (997119864 + 02)

+ NaN 0 834119864 + 02 (381119864 + 02) NaN 0

F6 158119864 minus 13 (782119864 minus 13)sim146119864 + 05 30 223119864 + 02 (803119864 + 01)

+ NaN 0 211119864 minus 09 (668119864 minus 09) 185119864 + 05 30


+ NaN 0 159119864 minus 02 (116119864 minus 02) 480119864 + 04 12



F9 185119864 + 02 (175119864 + 01)+ NaN 0 397119864 minus 02 (204119864 minus 02)

+ NaN 0 592119864 minus 17 (324119864 minus 16) 771119864 + 04 30

F10 205119864 + 02 (196119864 + 01)+ NaN 0 526119864 + 01 (119119864 + 01)

+ NaN 0 347119864 + 01 (124119864 + 01) NaN 0

F11 395119864 + 01 (103119864 + 00)+ NaN 0 309119864 + 01 (305119864 + 00)

+ NaN 0 172119864 + 01 (658119864 + 00) NaN 0

F12 119119864 + 03 (232119864 + 03)sim231119864 + 05 3 141119864 + 04 (984119864 + 03)

+ NaN 0 207119864 + 03 (292119864 + 03) NaN 0

F13 159119864 + 01 (112119864 + 00)+ NaN 0 109119864 + 00 (234119864 minus 01)


F14 134119864 + 01 (151119864 minus 01)+ NaN 0 131119864 + 01 (383119864 minus 01)

+ NaN 0 127119864 + 01 (240119864 minus 01) NaN 0

better 16 24

similar 9 0











respectively






10minus45

10minus40

10minus35

10minus30

10minus25

10minus20

10minus15

Fitn

ess v

alue

s (lo

g)

Sphere

10minus15

10minus10

10minus5

100

105

Schwefel

Local search rate

Fitn

ess v

alue

s (lo

g)

10minus30

10minus25

10minus20

10minus15

10minus10

Schwefel2

10minus35

10minus30

10minus25

10minus20


10minus28

10minus27

10minus26

10minus25

F1

104

105

106 F3


0 02 04 06 08 1







101321

101322

1005

1006

1007

1008

1009

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)Fi

tnes

s val

ues (

log)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

F8 F13



0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1


st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO



0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











DE BBO MLBBO



+ NaN 0 278119864 minus 31 (325119864 minus 31) 283119864 + 04 30


+ NaN 0 143119864 minus 21 (711119864 minus 22) 574119864 + 04 30

f3 231119864 minus 19 (237119864 minus 19)+151119864 + 05 30 126119864 + 02 (519119864 + 01)

+ NaN 0 190119864 minus 20 (303119864 minus 20) 140119864 + 05 30


+ NaN 0 449119864 minus 08 (125119864 minus 07) 349119864 + 05 30


+ NaN 0 334119864 minus 21 (908119864 minus 21) 225119864 + 05 30

f6 867119864 minus 01 (120119864 + 00)+251119864 + 04 14 177119864 + 00 (117119864 + 00)

+119119864 + 05 3 000119864 + 00 (000119864 + 00) 114119864 + 04 30



f8 729119864 + 03 (232119864 + 02)+ NaN 0 253119864 minus 01 (882119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 553119864 + 04 30

f9 176119864 + 02 (101119864 + 01)+ NaN 0 356119864 minus 02 (152119864 minus 02)

+ NaN 0 000119864 + 00 (000119864 + 00) 916119864 + 04 30


+ NaN 0 610119864 minus 15 (114119864 minus 15) 505119864 + 04 30


+ NaN 0 287119864 minus 03 (502119864 minus 03) 333119864 + 04 22


+ NaN 0 588119864 minus 28 (322119864 minus 27) 257119864 + 04 30


+ NaN 0 509119864 minus 32 (313119864 minus 32) 271119864 + 04 30


+ NaN 0 303119864 minus 29 (697119864 minus 29) 290119864 + 04 30

F2 759119864 minus 12 (842119864 minus 12)+172119864 + 05 30 355119864 + 03 (135119864 + 03)

+ NaN 0 197119864 minus 12 (255119864 minus 12) 158119864 + 05 30

F3 139119864 + 05 (106119864 + 05)+ NaN 0 114119864 + 07 (490119864 + 06)

+ NaN 0 926119864 + 04 (490119864 + 04) NaN 0

F4 710119864 minus 05 (161119864 minus 04)sim285119864 + 05 1 139119864 + 04 (456119864 + 03)

+ NaN 0 700119864 minus 05 (187119864 minus 04) 290119864 + 05 5

F5 531119864 + 01 (137119864 + 02)minus NaN 0 581119864 + 03 (997119864 + 02)

+ NaN 0 834119864 + 02 (381119864 + 02) NaN 0

F6 158119864 minus 13 (782119864 minus 13)sim146119864 + 05 30 223119864 + 02 (803119864 + 01)

+ NaN 0 211119864 minus 09 (668119864 minus 09) 185119864 + 05 30


+ NaN 0 159119864 minus 02 (116119864 minus 02) 480119864 + 04 12



F9 185119864 + 02 (175119864 + 01)+ NaN 0 397119864 minus 02 (204119864 minus 02)

+ NaN 0 592119864 minus 17 (324119864 minus 16) 771119864 + 04 30

F10 205119864 + 02 (196119864 + 01)+ NaN 0 526119864 + 01 (119119864 + 01)

+ NaN 0 347119864 + 01 (124119864 + 01) NaN 0

F11 395119864 + 01 (103119864 + 00)+ NaN 0 309119864 + 01 (305119864 + 00)

+ NaN 0 172119864 + 01 (658119864 + 00) NaN 0

F12 119119864 + 03 (232119864 + 03)sim231119864 + 05 3 141119864 + 04 (984119864 + 03)

+ NaN 0 207119864 + 03 (292119864 + 03) NaN 0

F13 159119864 + 01 (112119864 + 00)+ NaN 0 109119864 + 00 (234119864 minus 01)


F14 134119864 + 01 (151119864 minus 01)+ NaN 0 131119864 + 01 (383119864 minus 01)

+ NaN 0 127119864 + 01 (240119864 minus 01) NaN 0

better 16 24

similar 9 0











respectively






10minus45

10minus40

10minus35

10minus30

10minus25

10minus20

10minus15

Fitn

ess v

alue

s (lo

g)

Sphere

10minus15

10minus10

10minus5

100

105

Schwefel

Local search rate

Fitn

ess v

alue

s (lo

g)

10minus30

10minus25

10minus20

10minus15

10minus10

Schwefel2

10minus35

10minus30

10minus25

10minus20


10minus28

10minus27

10minus26

10minus25

F1

104

105

106 F3


0 02 04 06 08 1







101321

101322

1005

1006

1007

1008

1009

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)Fi

tnes

s val

ues (

log)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

F8 F13



0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1


st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO



0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














10minus45

10minus40

10minus35

10minus30

10minus25

10minus20

10minus15

Fitn

ess v

alue

s (lo

g)

Sphere

10minus15

10minus10

10minus5

100

105

Schwefel

Local search rate

Fitn

ess v

alue

s (lo

g)

10minus30

10minus25

10minus20

10minus15

10minus10

Schwefel2

10minus35

10minus30

10minus25

10minus20


10minus28

10minus27

10minus26

10minus25

F1

104

105

106 F3


0 02 04 06 08 1







101321

101322

1005

1006

1007

1008

1009

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)Fi

tnes

s val

ues (

log)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

Fitn

ess v

alue

s (lo

g)

F8 F13



0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1


st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO



0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ssSphere

0 05 1 15 2times10

5

10minus40

10minus20

100

1020

1040 Schwefel3


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus10

10minus5

100

105

Schwefel4


Best

fitne

ss

0 1 2 3 4 5times10

5

10minus30

10minus20

10minus10

100

1010 Rosenbrock


Best

fitne

ss

0 5 10 15times10

4

10minus2

100

102

104

106 Step


Best

fitne

ss

0 05 1 15 2 25times10

5

100

Quartic


Best

fitne

ss0 5 10 15

times104

10minus20

100

Penalized2


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus30

10minus20

10minus10

100

1010 F1


st fit

ness

0 05 1 15 2 25 3times10

5

104

106

108

1010 F3


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus5

100

105

F4


Best

fitne

ss

0 05 1 15 2 25 3times10

5

101

102

103

104

105 F5


Best

fitne

ss

0 05 1 15 2 25times10

5

10minus2

100

102

F7


Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus15

10minus10

10minus5

100

105 Schwefel


Best

fitne

ss

0 05 1 15 2 25 3times10

5

F8


101319

101322

101325

101328

Best

fitne

ss

0 05 1 15 2 25 3times10

5

10minus20

10minus10

100

1010 F9


Best

fitne

ss

0 05 1 15 2 25 3times10

5

1013

1014

1015

1016

F11


Best

fitne

ss

MLBBODE

BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO

0 05 1 15 2 25times10

5

100

102

F13

Best

fitne

ss


BBO



0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

Schwefel3

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

Schwefel4

MLBBO DE BBO

Best

fitne

ss

0

50

100

150Rosenbrock

MLBBO DE BBO

Best

fitne

ss

0

1

2

3

4

5Step

MLBBO DE BBO

Best

fitne

ss

02468

10times10

minus3 Quartic

MLBBO DE BBO

Best

fitne

ss0

2000

4000

6000

8000 Schwefel

MLBBO DE BBO

Best

fitne

ss

0001002003004005

Penalized2

MLBBO DE BBO

Best

fitne

ss

0

005

01

015

F1

MLBBO DE BBOBe

st fit

ness

005

115

225times10

7 F3

MLBBO DE BBO

Best

fitne

ss

005

115

225times10

4 F4

MLBBO DE BBO

Best

fitne

ss

0

2000

4000

6000

8000F5

MLBBO DE BBO

Best

fitne

ss

0

05

1

F7

MLBBO DE BBO

Best

fitne

ss

206

207

208

209

21

F8

MLBBO DE BBO

Best

fitne

ss

0

50

100

150

200F9

MLBBO DE BBO

Best

fitne

ss

0

02

04

06Be

st fit

ness

Sphere

MLBBO DE BBO

Best

fitne

ss

10

20

30

40F11

MLBBO DE BBO

Best

fitne

ss

0

5

10

15

F13

MLBBO DE BBO

Best

fitne

ss

12

125

13

135

F14

MLBBO DE BBO

Best

fitne

ss






+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













+170119864 minus 09 (930119864 minus 09)

sim226119864 minus 03 (106119864 minus 03)

+211119864 minus 31 (125119864 minus 31)


+274119864 minus 05 (148119864 minus 04)

sim896119864 minus 02 (158119864 minus 02)

+158119864 minus 21 (725119864 minus 22)


+325119864 minus 02 (450119864 minus 02)

+219119864 + 01 (934119864 + 00)

+142119864 minus 20 (170119864 minus 20)


+243119864 minus 02 (674119864 minus 03)

+120119864 + 00 (125119864 + 00)

+528119864 minus 08 (102119864 minus 07)

f5 365119864 + 01 (220119864 + 01)+360119864 + 01 (340119864 + 01)

+632119864 + 01 (833119864 + 01)

+425119864 + 01 (374119864 + 01)

+534119864 minus 21 (110119864 minus 20)

f6 000119864 + 00 (000119864 + 00)sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)





f8 240119864 + 02 (182119864 + 02)+170119864 minus 11 (226119864 minus 12)

+155119864 minus 11 (850119864 minus 11)

sim817119864 minus 05 (315119864 minus 05)

+000119864 + 00 (000119864 + 00)


sim545119864 minus 05 (299119864 minus 04)

sim138119864 minus 02 (511119864 minus 03)

+000119864 + 00 (000119864 + 00)


+974119864 minus 08 (597119864 minus 09)

+294119864 minus 02 (663119864 minus 03)

+610119864 minus 15 (649119864 minus 16)


sim508119864 minus 03 (105119864 minus 02)

sim792119864 minus 02 (436119864 minus 02)

+173119864 minus 03 (400119864 minus 03)


sim346119864 minus 03 (189119864 minus 02)

sim339119864 minus 05 (347119864 minus 05)

+235119864 minus 32 (398119864 minus 32)


sim633119864 minus 13 (344119864 minus 12)

sim643119864 minus 04 (966119864 minus 04)

+564119864 minus 32 (338119864 minus 32)


sim116119864 minus 07 (634119864 minus 07)

sim350119864 minus 04 (925119864 minus 05)

+202119864 minus 29 (616119864 minus 29)

F2 470119864 + 01 (864119864 + 00)+292119864 + 00 (202119864 + 00)

+266119864 + 00 (216119864 + 00)

+872119864 + 02 (443119864 + 02)

+137119864 minus 12 (153119864 minus 12)

F3 156119864 + 06 (345119864 + 05)+210119864 + 06 (686119864 + 05)

+237119864 + 06 (989119864 + 05)

+446119864 + 06 (158119864 + 06)

+902119864 + 04 (554119864 + 04)

F4 318119864 + 03 (994119864 + 02)+774119864 + 02 (386119864 + 02)

+146119864 + 01 (183119864 + 01)

+217119864 + 04 (850119864 + 03)

+175119864 minus 05 (292119864 minus 05)

F5 103119864 + 04 (163119864 + 03)+368119864 + 03 (627119864 + 02)

+504119864 + 03 (124119864 + 03)

+631119864 + 03 (118119864 + 03)

+800119864 + 02 (379119864 + 02)

F6 320119864 + 02 (964119864 + 02)sim871119864 + 02 (212119864 + 03)

+276119864 + 02 (846119864 + 02)

sim845119864 + 02 (269119864 + 03)

sim355119864 minus 09 (140119864 minus 08)


sim226119864 minus 02 (199119864 minus 02)

+764119864 minus 01 (139119864 + 00)

+138119864 minus 02 (121119864 minus 02)


minus207119864 + 01 (171119864 minus 01)

minus207119864 + 01 (851119864 minus 02)

minus209119864 + 01 (548119864 minus 02)


sim122119864 minus 07 (661119864 minus 07)

sim132119864 minus 02 (553119864 minus 03)

+592119864 minus 17 (324119864 minus 16)

F10 643119864 + 01 (281119864 + 01)+329119864 + 01 (880119864 + 00)

sim714119864 + 01 (206119864 + 01)

+480119864 + 01 (162119864 + 01)

+369119864 + 01 (145119864 + 01)

F11 236119864 + 01 (319119864 + 00)+209119864 + 01 (448119864 + 00)

sim270119864 + 01 (411119864 + 00)

+319119864 + 01 (390119864 + 00)

+195119864 + 01 (787119864 + 00)

F12 118119864 + 04 (866119864 + 03)+468119864 + 03 (444119864 + 03)

+504119864 + 03 (410119864 + 03)

+129119864 + 04 (894119864 + 03)

+243119864 + 03 (297119864 + 03)


minus109119864 + 00 (229119864 minus 01)


minus298119864 + 00 (162119864 minus 01)

F14 125119864 + 01 (361119864 minus 01)sim119119864 + 01 (617119864 minus 01)

minus133119864 + 01 (344119864 minus 01)

+131119864 + 01 (339119864 minus 01)

+127119864 + 01 (239119864 minus 01)


































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 5 Continued









MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

MLBBOOBBOPBBO

MOBBORCBBO

0 05 1 15 2 25 3 3510

13

1014

1015

1016

Best

fitne

ss

F11

times105


05 1 15 2 25 3 3510

minus2

100

102

F7

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus30

10minus20

10minus10

100

1010 F1

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Rastrigin

times105


Best

fitne

ss

0 5 10 15

10minus15

10minus10

10minus5

100

Ackley

times104


Best

fitne

ss

0 05 1 15 2 2510

minus4

10minus2

100

102

104 Griewank

times105


Best

fitne

ss

0 05 1 15 2 25 3 3510

minus15

10minus10

10minus5

100

105

Schwefel

times105


st fit

ness

0 1 2 3 4 5 610

minus8

10minus6

10minus4

10minus2

100

102

Schwefel4

times105


Best

fitne

ss

0 05 1 15 2times10

5

10minus40

10minus30

10minus20

10minus10

100

1010


Sphere

Best

fitne

ss



0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6times10

5

10minus20

10minus10

100

1010


Best

fitne

ss

Schwefel2

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

MLBBOOXDEDEBBO

0 1 2 3 4 5 6times10

5


10minus30

10minus20

10minus10

100

1010

Rosenbrock

Best

fitne

ss

0 2 4 6 8 1010

minus2

100

102

104

106

Step

times104


Best

fitne

ss

0 05 1 15 210

minus40

10minus20

100

1020

Penalized1

times105


Best

fitne

ss

0 1 2 3 410

4

106

108

1010

F3

Best

fitne

ss

times105


102

103

104

105

F5

times105


Best

fitne

ss

10minus20

10minus10

100

1010

F9

0 1 2 3 4times10

5


Best

fitne

ss

101

102

103

F10

0 1 2 3 4times10

5


Best

fitne

ss

0 1 2 3

10111

10113

10115

F14

times105


Best

fitne

ss






0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10times10

5

10minus50

100

1050

10100

Schwefel3

Best

fitne

ss

(1) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus10

10minus5

100

105

Schwefel

times105

Best

fitne

ss

(2) NFFEs (Dim = 100)

0 5 1010

minus40

10minus20

100

1020

Penalized1

times105

Best

fitne

ss

(3) NFFEs (Dim = 100)

0 05 1 15 210

0

102

104

106

Schwefel2

times106

Best

fitne

ss

(4) NFFEs (Dim = 200)0 05 1 15 2

10minus20

10minus10

100

1010

Ackley

times106

Best

fitne

ss

(5) NFFEs (Dim = 200)0 05 1 15 2

10minus40

10minus20

100

1020

Penalized2

times106

Best

fitne

ss

(6) NFFEs (Dim = 200)

0 2 4 6 8 1010

minus40

10minus20

100

1020 F1

times104

Best

fitne

ss

(7) NFFEs (Dim = 10)0 2 4 6 8 10

10minus15

10minus10

10minus5

100

105 F5

times104

Best

fitne

ss

(8) NFFEs (Dim = 10)0 2 4 6 8 10

10131

10132

F8

times104

Best

fitne


0 2 4 6 8 1010

minus1

100

101

102

103 F13

times104

Best

fitne

ss

(10) NFFEs (Dim = 10)0 1 2 3 4 5

105

1010 F3

times105

Best

fitne

ss

(11) NFFEs (Dim = 50)

0 1 2 3 4 510

1

102

103

104 F10

times105

Best

fitne

ss

(12) NFFEs (Dim = 50)

0 1 2 3 4 5

1016

1017

1018

1019

F11

times105

Best

fitne

ss

(13) NFFEs (Dim = 50)0 1 2 3 4 5

103

104

105

106

107

F12

times105

Best

fitne

ss

(14) NFFEs (Dim = 50)0 2 4 6 8 10

10minus5

100

105

1010

F2

times105

Best

fitne

ss

(15) NFFEs (Dim = 100)

0 2 4 6 8 10106

108

1010

1012

F3

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(16) NFFEs (Dim = 100)

0 2 4 6 8 10104

105

106

107

F4

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(17) NFFEs (Dim = 100)

0 2 4 6 8 1010

minus15

10minus10

10minus5

100

105 F9

MLBBODE

BBODEBBO

Best

fitne

ss

times105

(18) NFFEs (Dim = 100)



0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 2000

2

4

6

8

10

12

14

16

18times10

5

NFF

Es

f1f2f3f4f5f6f7

f8f9f10f11f12f13

(a) Dimension

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

NFF

Es

F1F2F4

F6F7F9

times105

(b) Dimension


0 5 10 15times10

4

10minus40

10minus20

100

1020


Best

fitne

ss

Sphere

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

MLBBOMLBBO2

MLBBO3MLBBO4

10minus2

100

102

104

106

Step

0 1 2 3times10

4


Best

fitne

ss

10minus15

10minus10

10minus5

100

105

Rastrigin

0 1 2 3times10

5


Best

fitne

ss

1014

1015

1016

F11

0 1 2 3times10

5


Best

fitne

ss

104

106

108

1010 F3

0 1 2 3times10

5


Best

fitne

ss

10minus40

10minus20

100

1020 F1

0 1 2 3times10

5


Best

fitne

ss





dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












dim = 10F1 000119864 + 00 (000119864 + 00) 471119864 minus 29 (806119864 minus 29)

+186119864 minus 02 (976119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim000119864 + 00 (000119864 + 00)


+850119864 minus 25 (282119864 minus 24)

sim365119864 minus 23(258119864 minus 22)


sim301119864 + 02 (331119864 + 02)

+227119864 minus 21 (378119864 minus 21)

+196119864 minus 02 (619119864 minus 02)


+288119864 minus 05 (148119864 minus 04)

+408119864 + 02 (693119864 + 02)


+410119864 + 00 (220119864 + 00)

+140119864 + 02 (822119864 + 02)


+386119864 minus 02 (236119864 minus 02)

minus115119864 + 00 (108119864 + 00)


sim204119864 + 01 (513119864 minus 02)

sim203119864 + 01 (835119864 minus 02)

F9 000119864 + 00 (000119864 + 00) 839119864 + 00 (694119864 + 00)+888119864 minus 03 (451119864 minus 03)

+000119864 + 00 (000119864 + 00)

sim312119864 minus 09 (221119864 minus 08)

F10 616119864 + 00 (297119864 + 00) 229119864 + 01 (574119864 + 00)+182119864 + 01 (869119864 + 00)

+598119864 + 00 (242119864 + 00)

sim157119864 + 01 (604119864 + 00)

F11 172119864 + 00 (159119864 + 00) 248119864 + 00 (386119864 + 00)sim693119864 + 00 (143119864 + 00)

+828119864 minus 01 (139119864 + 00)

minus mdashF12 347119864 + 02 (608119864 + 02) 255119864 + 02 (530119864 + 02)

sim723119864 + 02 (748119864 + 02)

+104119864 + 02 (340119864 + 02)


+362119864 minus 01 (135119864 minus 01)



sim359119864 + 00 (382119864 minus 01)

+309119864 + 00 (176119864 minus 01)

+ mdashdim = 50


+000119864 + 00 (000119864 + 00)

minus000119864 + 00 (000119864 + 00)


+104119864 + 03 (304119864 + 02)

+262119864 minus 02 (717119864 minus 02)

F3 186119864 + 05 (684119864 + 04) 549119864 + 05 (201119864 + 05)+193119864 + 07 (590119864 + 06)

+720119864 + 06 (238119864 + 06)

+133119864 + 06 (489119864 + 05)

F4 173119864 + 01 (158119864 + 01) 319119864 + 02 (252119864 + 02)+436119864 + 04 (119119864 + 04)

+726119864 + 03 (217119864 + 03)

+129119864 + 04 (715119864 + 03)

F5 418119864 + 03 (624119864 + 02) 266119864 + 03 (509119864 + 02)minus116119864 + 04 (163119864 + 03)

+288119864 + 03 (422119864 + 02)

minus140119864 + 04 (267119864 + 03)

F6 288119864 + 00 (148119864 + 01) 116119864 + 00 (183119864 + 00)sim305119864 + 02 (839119864 + 01)

+536119864 + 01 (214119864 + 01)

+406119864 + 01 (566119864 + 01)


+279119864 minus 03 (655119864 minus 03)



minus211119864 + 01 (421119864 minus 02)

sim211119864 + 01 (539119864 minus 02)


+332119864 minus 02 (182119864 minus 01)

sim222119864 minus 08 (157119864 minus 07)

F10 866119864 + 01 (222119864 + 01) 430119864 + 02 (336119864 + 01)+998119864 + 01 (282119864 + 01)

+163119864 + 02 (135119864 + 01)

+148119864 + 02 (415119864 + 01)

F11 360119864 + 01 (840119864 + 00) 728119864 + 01 (186119864 + 00)+585119864 + 01 (462119864 + 00)

+592119864 + 01 (144119864 + 00)

+ mdashF12 677119864 + 03 (532119864 + 03) 737119864 + 03 (805119864 + 03)

sim515119864 + 04 (261119864 + 04)

+105119864 + 04 (938119864 + 03)

sim mdashF13 556119864 + 00 (276119864 minus 01) 325119864 + 01 (206119864 + 00)

+186119864 + 00 (258119864 minus 01)

minus523119864 + 00 (288119864 minus 01)


+227119864 + 01 (376119864 minus 01)

+226119864 + 01 (162119864 minus 01)

+ mdashdim = 100


+168119864 minus 29 (519119864 minus 29)


+398119864 + 04 (734119864 + 03)

+242119864 + 04 (508119864 + 03)

+ mdashF3 202119864 + 06 (535119864 + 05) 206119864 + 06 (713119864 + 05)

sim452119864 + 07 (983119864 + 06)

+142119864 + 07 (415119864 + 06)

+ mdashF4 163119864 + 04 (827119864 + 03) 431119864 + 04 (113119864 + 04)

+148119864 + 05 (252119864 + 04)

+847119864 + 04 (116119864 + 04)

+ mdashF5 132119864 + 04 (140119864 + 03) 811119864 + 03 (129119864 + 03)

minus301119864 + 04 (332119864 + 03)

+608119864 + 03 (628119864 + 02)

minus mdashF6 398119864 + 01 (359119864 + 01) 124119864 + 02 (505119864 + 01)

+474119864 + 02 (532119864 + 01)

+115119864 + 02 (281119864 + 01)


sim121119864 + 00 (638119864 minus 02)

+148119864 minus 03 (397119864 minus 03)


sim211119864 + 01 (563119864 minus 02)

minus213119864 + 01 (205119864 minus 02)


+134119864 minus 01 (310119864 minus 02)

+156119864 + 00 (110119864 + 00)

+ mdashF10 296119864 + 02 (514119864 + 01) 110119864 + 03 (450119864 + 01)

+274119864 + 02 (377119864 + 01)

sim397119864 + 02 (250119864 + 01)








+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













+ 0 217119864 minus 04 (792119864 minus 05)+ 0


+ 0 364119864 minus 02 (700119864 minus 03)+ 0


+ 0 198119864 + 00 (563119864 minus 01)+ 0


+ 0 232119864 minus 02 (647119864 minus 03)+ 0


+ 0 364119864 + 01 (358119864 + 01)+ 0

f6 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 000119864 + 00 (000119864 + 00)

sim 30 000119864 + 00 (000119864 + 00)sim 30


sim 30 128119864 minus 02 (507119864 minus 03)+ 11

f8 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 163119864 minus 06 (125119864 minus 06)

+ 6 792119864 minus 06 (284119864 minus 06)+ 0

f9 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 793119864 minus 04 (464119864 minus 04)

+ 0 360119864 minus 03 (146119864 minus 03)+ 0


+ 0 972119864 minus 03 (150119864 minus 03)+ 0


+ 1 816119864 minus 02 (571119864 minus 02)+ 0


sim 26 768119864 minus 06 (128119864 minus 05)+ 1


sim 0 570119864 minus 05 (507119864 minus 05)+ 0

F1 000119864 + 00 (000119864 + 00) 30 000119864 + 00 (000119864 + 00)sim 30 948119864 minus 06 (828119864 minus 06)

+ 2 466119864 minus 05 (191119864 minus 05)+ 0


+ 0 239119864 + 01 (101119864 + 01)+ 0

F3 948119864 + 04 (514119864 + 04) 0 186119864 + 05 (984119864 + 04)+ 0 186119864 + 06 (575119864 + 05)

+ 0 951119864 + 06 (757119864 + 06)+ 0


+ 0 489119864 + 03 (177119864 + 03)+ 0

F5 818119864 + 02 (325119864 + 02) 0 127119864 + 02 (659119864 + 01)minus 0 609119864 + 03 (112119864 + 03)

+ 0 447119864 + 03 (700119864 + 02)+ 0


+ 0 209119864 + 03 (448119864 + 03)+ 0


+ 0 776119864 minus 01 (139119864 + 00)+ 0


sim 0 210119864 + 01 (380119864 minus 02)sim 0


+ 30 344119864 minus 03 (145119864 minus 03)+ 30

F10 376119864 + 01 (140119864 + 01) 0 974119864 + 01 (654119864 + 00)+ 0 379119864 + 01 (105119864 + 01)

sim 0 122119864 + 02 (819119864 + 00)+ 0

F11 210119864 + 01 (737119864 + 00) 0 321119864 + 01 (105119864 + 00)+ 0 263119864 + 01 (496119864 + 00)

+ 0 334119864 + 01 (107119864 + 00)+ 0

F12 207119864 + 03 (271119864 + 03) 0 136119864 + 04 (197119864 + 04)+ 0 126119864 + 04 (929119864 + 03)

+ 0 805119864 + 03 (703119864 + 03)+ 0




minus 0 130119864 + 01 (186119864 minus 01)+ 0







6 Appendix

See Table 8


Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table8Be

nchm

arkfunctio

nsused

inou

rexp

erim

entaltests

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f11198911(119909)=

119899

sum 119894=1

1199092 119894

Sphere

n[minus100100]

0

f21198912(119909)=

119899

sum 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816+

119899

prod 119894=1

1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816Schw

efelrsquosP

roblem

222

n[1010]

0

f31198913(119909)=

119899

sum 119894=1

(

119894

sum 119895=1

119909119895)2

Schw

efelrsquosP

roblem

12n

[minus100100]

0

f41198914(119909)=max1198941003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 10038161le119894le119899

Schw

efelrsquosP

roblem

221

n[minus100100]

0

f51198915(119909)=

119899minus1

sum 119894=1

[100(119909119894+1minus1199092 119894)2+(119909119894minus1)2]

Rosenb

rock

functio

nn

[minus3030]

0

f61198916(119909)=

119899

sum 119894=1


Step

functio

nn

[minus100100]

0

f71198917(119909)=

119899

sum 119894=1

1198941199094 119894+rand

om[01)

Quarticfunctio

nn

[minus12

812

8]0

f81198918(119909)=minus

119899

sum 119894=1

119909119894sin(radic1003816 1003816 1003816 1003816119909119894

1003816 1003816 1003816 1003816)Schw

efelfunctio

nn

[minus512512]

minus125695

f91198919(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

Rastrig

infunctio

nn

[minus512512]

0

f10


119899

sum 119894=1

1199092 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119909119894)+20+119890

Ackley

functio

nn

[minus3232]

0

f11

11989111(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos (119909119894

radic119894

)+1

Grie

wankfunctio

nn

[minus60

060

0]0

f12

11989112(119909)=120587 11989910sin2(120587119910119894)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]

+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+(119909119894minus1)

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)2

0 119896(minus119909119894minus119886)2

119909119894gt119886

minus119886le119909119894le119886

119909119894ltminus119886

Penalized

functio

n1

n[minus5050]

0


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

f13

11989113(119909)=01sin2(31205871199091)+

119899minus1

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]

+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

Penalized

functio

n2

n[minus5050]

0

F11198651=

119899

sum 119894=1

1199112 119894+119891

bias1119885=119883minus119874

Shifted

sphere

functio

nn

[minus100100]

minus450

F21198652=

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)2+119891

bias2119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12n

[minus100100]

minus450

F31198653=

119899

sum 119894=1

(106)(119894minus1)(119863minus1)1199112 119894+119891

bias3119885=119883minus119874

Shifted

rotatedhigh

cond

ition

edellip

ticfunctio

nn

[minus100100]

minus450

F41198654=(

119899

sum 119894=1

(

119894

sum 119895=1

119911119895)

2

)lowast(1+04|119873(01)|)+119891

bias4119885=119883minus119874

Shifted

Schw

efelrsquosP

roblem

12with

noise

infitness

n[minus100100]

minus450

F51198655=max1003816 1003816 1003816 1003816119860119894119909minus119861119894

1003816 1003816 1003816 1003816+119891

bias5

Schw

efelrsquosP

roblem

26with

glob

alop

timum

onbo

unds

n[minus3030]

minus310

F61198656=

119899minus1

sum 119894=1

[100(119911119894+1minus1199112 119894)2+(119911119894minus1)2]+119891

bias6119885=119883minus119874+1

Shifted

rosenb

rockrsquos

functio

nn

[minus100100]

390


Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table8Con

tinued

Functio

nsMathematicalform

ula

Nam

eDim

ensio

nSearch

space

Optim

a

F71198657=

1

4000

119899

sum 119894=1

1199112 119894minus

119899

prod 119894=1

cos(119911119894

radic119894

)+1+119891


Shifted

rotatedGrie

wankrsquos

functio

nwith

outb

ound

sn

Outsid

eof

initializa-

tionrange

[060

0]

minus180


119899

sum 119894=1

1199112 119894)minusexp(1 119899

119899

sum 119894=1

cos2120587119911119894)+20+119890+119891


Shifted

rotatedAc

kleyrsquos

functio

nwith

glob

alop

timum

onbo

unds

n[minus3232]

minus140

F91198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891

bias8119885=119883minus119874

Shifted

Rastrig

inrsquosfunctio

nn

[minus512512]

minus330

F10

1198659=

119899

sum 119894=1

[1199112 119894minus10cos(2120587119911119894)+10]+119891


Shifted

rotatedRa

strig

inrsquos

functio

nn

[minus512512]

minus330

F11

11986511=

119899

sum 119894=1

(

119896maxsum 119896=0

[119886119896cos(2120587119887119896(119911119894+05))])minus119899

119896maxsum 119896=0

[119886119896cos(2120587119887119896sdot05)]+119891

bias11

119886=05119887=3119896max=20119885=(119883minus119874)lowast119872

Shifted

rotatedweierstrass

Functio

nn

[minus60

060

0]90

F12

11986512=

119899

sum 119894=1

(119860119894minus119861119894(119909))

2

+119891

bias12

119860119894=

119899

sum 119894=1

(119886119894119895sin120572119895+119887119894119895cos120572119895)119861119894=

119899

sum 119894=1

(119886119894119895sin119909119895+119887119894119895cos119909119895)

Schw

efelrsquosP

roblem

213

n[minus100100]

minus46

0

F13


bias13

119885=119883minus119874+1

Expand

edextend

edGrie

wankrsquos

plus

Rosenb

rockrsquosfunctio

n(F8F

2)

n[minus5050]

minus130

F14

119865(119909119910)=05+

sin2(radic1199092+1199102)minus05

(1+0001(1199092+1199102))2



Shifted

rotatedexpand

edScafferrsquosF6

n[minus100100]

minus300




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Modified Biogeography-Based …downloads.hindawi.com/journals/jam/2013/960524.pdfResearch Article Modified Biogeography-Based Optimization with Local Search Mechanism