Research Article Cloud Particles Evolution Algorithmdownloads.hindawi.com/journals/mpe/2015/434831.pdf · 2019. 7. 31. · search probabilistically in the solution space to create

Research ArticleCloud Particles Evolution Algorithm

Wei Li Lei Wang Qiaoyong Jiang Xinhong Hei and Bin Wang

School of Computer Science and Engineering Xirsquoan University of Technology Xirsquoan 710048 China

Correspondence should be addressed to Lei Wang wangleeei163com

Received 10 November 2014 Revised 13 January 2015 Accepted 13 January 2015

Academic Editor Jyh-Hong Chou

Copyright copy 2015 Wei Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Many evolutionary algorithms have been paid attention to by the researchers and have been applied to solve optimization problemsThis paper presents a new optimization method called cloud particles evolution algorithm (CPEA) to solve optimization problemsbased on cloud formation process and phase transformation of natural substance The cloud is assumed to have three states inthe proposed algorithm Gaseous state represents the global exploration Liquid state represents the intermediate process fromthe global exploration to the local exploitation Solid state represents the local exploitation The cloud is composed of descript andindependent particles in this algorithmThe cloud particles use phase transformation of three states to realize the global explorationand the local exploitation in the optimization process Moreover the cloud particles not only realize the survival of the fittestthrough competition mechanism but also ensure the diversity of the cloud particles by reciprocity mechanismThe effectiveness ofthe algorithm is validated upon different benchmark problems The proposed algorithm is compared with a number of other well-known optimization algorithms and the experimental results show that cloud particles evolution algorithm has a higher efficiencythan some other algorithms

1 Introduction

Many real-world problems which are classified as global opti-mization problems are very complex and are quite difficult tosolve Various optimization algorithms are developed to solvethese problems Most of these algorithms learn and imitatea variety of intelligent behavior in nature These algorithmsexploit a set of potential solutions and detect the optimalsolution through cooperation and competition among theindividuals of the population [1] For example Genetic algo-rithm [2] proposed by Professor Holland is a computationalmodel of biological evolutionwhich imitates natural selectionand genetic mechanism Ant colony algorithm [3] is a heuris-tic bionic optimization algorithmwhich imitates the foragingbehavior of the ant colony In the ant colony algorithman individual is called an artificial ant The artificial antssearch probabilistically in the solution space to create can-didate solutions The evaluating and the updating of thesecandidate solutions are based on the pheromone associatedwith each one of them The candidate solution with themaximum amount of pheromone is considered to be theoptimal solution of the problem Particle swarm optimization(PSO) [4] is a bionic intelligent computing method proposed

by Kennedy and Eberhart which imitates the flying and theforaging behavior of birds In PSO the optimal solution iscalled a leader The leader guides each particle to searchBesides the particles cooperate with each other to realizeexploration These behaviors are directed by a combinationof swarmrsquos previous best (gBest) and their own previous best(pBest) [5] The artificial bee colony algorithm (ABC) [6 7]is a nonderivative optimization method based on self-organ-izational models of bee and swarm intelligence In ABC beesare divided into three groups which are called ldquoemployedrdquobees ldquoonlookerrdquo bees and ldquoscoutsrdquo bees ldquoOnlookersrdquo beeslook for food sources Food source is the search goal of beesldquoEmployedrdquo bees work on food source In addition ldquoscoutsrdquobees search for the new food source near the hive Artificialimmune system (AIS) [8] is an adaptive system inspired bythe immunologywhich imitates human immunological func-tions principles and models to solve complex problems InAIS the external organisms are called antigen and the inter-nal organisms are called antibodies Immune clonal oper-ation provides multistrategy conditions for recombinationand mutation [9] The clonal selection selects the excellentantibodies from the subpopulation generated by clonal pro-liferation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 434831 21 pageshttpdxdoiorg1011552015434831

2 Mathematical Problems in Engineering

This paper which is inspired by the natural phenomenaof cloud formation presents a novel optimization algorithmto solve optimization problems called cloud particles evolu-tion algorithm (CPEA) The proposed algorithm uses phasetransformation mechanism to imitate the formation anddiversification of the cloud to solve the optimization prob-lems It is well known that the species evolution is influencedby individual fitness living environment competition andcooperation between the individuals Therefore the algo-rithm introduces competition and reciprocal mechanism inthe evolution process so as to strengthen the viability of thecloud particlesThe performance of the cloud particles evolu-tion algorithm (CPEA) is tested on the classical optimizationproblems The results are compared with other algorithmsin terms of best function value and the number of functionevaluations

The remainder of this paper is organized as follows InSection 2 the proposed CPEA and the concepts are intro-duced in detail In Section 3 the performance of the proposedalgorithm is validated by different optimization problemsThe conclusion and future research are drawn in Section 4

2 Cloud Particles Evolution Algorithm

21 Cloud Formation The idea of the proposed CPEA isinspired from cloud formation The state of the cloud parti-clesrsquo changing process [10ndash12] is shown in Figure 1

After the wet air rises it will go through a set of changesbecause of some certain reasons temperature and pressureThen the water vapor is formedWater vapor plays an impor-tant role in this process When the water vapor is saturatedit will adsorb on the cloud condensation nuclei (CNN) andthe initial cloud particles are formed Cloud particles arecondensed and sublimated continuously by absorbing watervapor around the cloud In the process of the condensationthe cloud particles are getting closer and colliding Then thelarger cloud particles are formed Ice phase is generated whenthe temperature is below 0∘C at this time The emergence ofice crystals damages the stable status of cloud phase structureIce crystals grow up quickly to be the snow crystals becauseof the condensation of the water vapor On the one handwater vapor transfers to ice crystals and ice crystals beginto grow up as supercooled water droplets evaporate On theother hand the supercooledwater dropletswill be frozen afterthe colliding to make the snow particles grow bigger If thesupercooledwater vapor which has been collided is toomanythe snow crystals will be transferred to be solid substancesThen the solid substances such as snow and hail will bedropped If the solid substances descent in the area which hasa higher temperature over 0∘C they will melt into raindropThewater in our planet will be irradiated by the sunshine andit will become the water vapor again in the air The cloud isformed and changed like this

22 Phase and Phase Transformation There are various kindsof substance in nature Substances commonly exist in threestates solid liquid and gaseous Figure 2 is a simple diagramto show howwater transforms among the three states Evapo-ration condensation andmelting are the threeways for water

Raindroplets

Cloudparticles

Snowpellets

cloudparticles

Large

Figure 1 Cloud particles transformation process

Depos

ition

Subli

mati

on

Melting

Freezing

Evap

orati

onCon

dens

ation

Figure 2 Three phases of water

transforming In order to describe the different forms of thesubstance the phase can be used as a symbol for states of sub-stances In thermodynamics phase is defined as some partswhich have the same quality [13] A phase transformation isthe transition which is from one phase to another in thermo-dynamics

Phase transformation is relevant to Gibbs free energyIn thermodynamics the Gibbs free energy [14] is a ther-modynamic potential that measures the ldquousefulrdquo or process-initiating work obtainable from a thermodynamic system ata constant temperature and pressure

23 Population Evolution Mechanism Darwin pointed outthat mutation and natural selection are the two importantfactors for the evolution which explained many natural phe-nomena successfully [15] Nowak from Harvard University

Mathematical Problems in Engineering 3

1 21

3

21

3

2

4

65

78

Cooperators

Kin selection Direct reciprocity Indirect reciprocity Network reciprocity Group selection

Defectors

1 r

Figure 3 Five mechanisms of cooperation evolution [16]

ranked the cooperation as the third important factor of thespecies evolution Evolution is a unity of the cooperationand the opposition [16] Every gene cell and organism ofthe individual should strengthen its own evolution process atthe expense of beating its rivals Therefore individuals oftenconflict with each other because of limited resources [17]However the complex life organizational structure in differ-ent levels is produced in the process of evolution which indi-cates that the cooperation exists in the evolution ThereforeNowak summarizes five mechanisms for the cooperationevolution such as kin selection direct reciprocity indirectreciprocity network reciprocity and group selection (shownin Figure 3)

Haldane believes that the altruism exists in the evolution[18] According to the study of the social insect Hamilton[19] used mathematic model for quantitative description anddefined the Hamilton rules The rule states natural selectionfavors cooperation if the donor and the recipient have kinrelationships Direct reciprocity requires several repeatedencounters between two individuals which can offer helpwith each other The direct reciprocity theory framework isrepeated PrisonerrsquosDilemmaThe classical game strategy is titfor tat TFT [20] The interactions among individuals innature are often asymmetrical and fleeting Therefore indi-rect reciprocity is a more prevalent reciprocity form Indirectreciprocity refers to establishing a good reputation by helpingothers which will be rewarded by others later In Indirectreciprocity one individual acts as donor while the other actsas recipient Indirect reciprocity will favor the cooperationevolution if 119902 the probability of one individualrsquos reputationexceeds the cost-to-benefit ratio of the altruistic act namely119902 gt 119888119887 [21] Network reciprocity is a new form of reciprocityCooperators can form network clusters in order to help eachother The cooperation mechanism of this spatial reciprocityis called network reciprocity Group selection is a minimaliststochastic model [22] The population is divided into severalgroups Cooperators help each other in their own groupswhile defectors do not helpThe pure cooperator groups growfaster than the pure defector groups while the defector groupsreproduce faster than the cooperator groups in any mixedgroup Therefore defectors are often selected in the lowerlevel (within groups) while cooperators are often selected inthe higher (between groups) level

0 2 4 6 8 100

2

4

6

8

10Gaseous phase

Cloud particle

Figure 4 Cloud gaseous phase

24 Description of the Proposed Algorithm The main goal ofthe cloud particles evolution system is to realize the evolutionof the cloud particles through phase transformation In cloudparticles evolution model cloud particles have three phasescloud gaseous phase cloud liquid phase and cloud solidphase Cloud gaseous phase indicates gaseous cloud particleswhich are distributed in the entire search space Cloud liquidphase refers to the rain in nature Cloud solid phase representshailstones in nature

The proposed algorithm which is similar to other meta-heuristic algorithms begins with an initial population calledthe cloud particles At the beginning let the initial state becloud gaseous phase which is composed of many cloud parti-cles (shown in Figure 4) The best individual (best cloud par-ticle) is chosen as a seed or nucleus Then the cloud particlesare going to have a condensational growth or collisions andcoalescence growth (shown in Figure 5) If the populationevolves successfully the cloud particles are going to changefrom cloud gaseous state to liquid state by condensationCloud changes from gaseous phase to cloud liquid phaseThisprocess is called phase transformation When cloud particlesare in liquid state cloud particles transform from globalexploration to local exploitation gradually In this processaccording to the idea of the indirect reciprocity mentionedbefore if 119902 gt 119888119887 the algorithm will favor the cooperationevolution or it will change from the liquid phase to solid phaseby phase transformation When cloud particles are in solid


0 2 4 6 8 100

2

4

6

8

10Liquid phase

Collision coalescence

Condensation sublimation

Figure 5 Cloud liquid phase

2 3 4 5 6 7 83

4

5

6

7

8Solid phase

Hailstones

Hailstones

Figure 6 Cloud solid phase

state (shown in Figure 6) the cloud particles condense intoice crystals at the beginning and then ice crystals curdle intohailstones The hailstone is the best solution

It can be seen from the biological evolutionary behaviorthat the individuals compete to survive However competi-tive ability is always asymmetry because the different individ-ual has different fitness Usually it is easier for disadvantagedindividuals to compete with the individuals in the same sit-uation while they seldom compete with the advantaged indi-viduals [23] Therefore competition and cooperation whichare two different survival strategies for living organisms areformed in order to adapt to the environment In the CPEAmodel according to the natural phenomena that the pop-ulation develops cooperatively when they evolve these twostrategies can be shown as followsThe populations carry outthe competitive evolution in cloud gaseous phase In cloudliquid phase the populations employ competitivemechanismfor improving local exploitation ability and employ reciprocalmechanism to ensure the diversity of the population Cloudsolid phase indicates the algorithm has found the optimalsolution area therefore the algorithm realizes fast conver-gence by competitive evolution The optimization process ofCPEA model is shown in Figure 7

In CPEA the initialization phase transformation drivingforce condensation operation reciprocity operation andsolidification operation are described as follows

Liquidphase

Solidphase

Optimizationsolution

Globalexploration

Conversionstate Local

exploitation

Gaseousphase

Reciprocalevolution

GlobalLocal

Competitiveevolution

Successfulevolution

Successfulevolution

Successful evolution

Phasetransition

Phase transition

Figure 7 Optimization process of CPEA model

241 Initialization When the metaheuristic methods areused to solve the optimization problem one population isformed at first Then the variables which are involved in theoptimization problem can be represented as an array In thisalgorithm one single solution is called one ldquocloud particlerdquoThe cloud particles have three parameters [24] They areExpectation (119864

119909) Entropy (119864

119899) and Hyperentropy (119867

119890) 119864119909

is the expectation values of the distribution for all cloud par-ticles in the domain 119864

119899is the range of domain which can be

accepted by linguistic values (qualitative concept) In anotherword 119864

119899is ambiguity119867

119890is the dispersion degree of entropy

(119864119899) That is to say119867

119890is the entropy of entropy In addition

119867119890can be also defined as the uncertaintymeasure for entropy

In 119898 dimensional optimization problems a cloud particle isan array of 1 times 119898 This array is defined as follows

Cloud particle = [1199091 1199092 119909

119898] (1)

To start the optimization algorithm amatrix of cloud par-ticles which size is119873times119898 is generated by cloud generator (iepopulation of cloud particles) Cloud particles generation isdescribed as follows Φ(120583 120575) is the normal random variablewhich has an expectation 120583 and a variance 120575119873 is the numberof cloud particle 119883 is the cloud particle generated by cloudgenerator Consider

119878 = 119904119894| Φ (119864

119899 119867119890) 119894 = 1 sdot sdot sdot 119873

119883 = 119909119894| Φ (119864

119909 119904119894) 119904119894isin 119878 119894 = 1 sdot sdot sdot 119873

(2)

The matrix 119883 generated by cloud generator is given as(rows and column are the number of population and thenumber of design variables resp)

Population of cloud particles

=

[[[[[[

[

Cloudparticle1

Cloudparticle2

Cloudparticle119873

]]]]]]

]

=

[[[[[[[

[

1198831

1

1198831

2

1198831

3

sdot sdot sdot 1198831

119898

1198832

1

1198832

2

1198832

3


119898

119883119873

1

119883119873

2

119883119873

3


119898

]]]]]]]

]

(3)


Each of the decision variable values (1199091 1199092 119909

119898) can

represent real values for continuous or discrete problemsThefitness of a cloud particle is calculated by the evaluation ofobjective function (119865) given as

119865119894= 119891 (119909

1

1

1199091

2

1199091

119894

) 119894 = 1 2 3 119898 (4)

In the function above119873 and 119898 are the number of cloudparticles and the number of decision variablesThe cloud par-ticles which number is119873 are generated by the cloud genera-torThe cloud particle which has the optimal value is selectedas the nucleus The total population consists of119873

119901subpopu-

lations Each subpopulation of cloud particles selects a cloudparticle as the nucleus Therefore there are119873

119901nuclei

242 Phase Transformation Driving Force The phase trans-formation driving force (PT) can reflect the evolution extentof the population In the proposed algorithm phase trans-formation driving force is the factor of deciding phase trans-form During the evolution the energy possessed by a cloudparticle is called its evolution potential Phase transformationdriving force is the evolution potential energy differencebetween the cloud particles of new generation and the cloudparticles of old generationThe phase transformation drivingforce is defined as follows

EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897

119894119894 = 1 2 3 119892119890119899 (5)

PT = radic(EPgenindexminus1 minus EP)

2

+ (EPgenindex minus EP)2

2

EP =EPgenindexminus1 + EPgenindex

2

(6)

The cloud evolution system realizes phase transformationif PT le 120575 EP represents the evolution potential of cloudparticles gen represents the maximum generation of thealgorithm bestindividual represents the best solution of thecurrent generation genindex is the number of the currentevolution

243 Condensation Operation The population realizes theglobal exploration in cloud gaseous phase by condensationoperation The population takes the best individual as thenucleus in the process of the condensation operation Inthe cloud gaseous phase the condensation growth space ofcloud particle is calculated by the following equations cd isthe condensation factor generation is the number of currentevolution 119864

119899119894is the entropy of the 119894th subpopulation 119867

119890119894is

hyperentropy of the 119894th subpopulation Consider

119864119899119894=

10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816

radic119892119890119899119890119903119886119905119894119900119899 (7)

119867119890119894=119864119899119894

cd (8)

244 Reciprocity Operation If the evolution potential ofcloud particles between the two generations is less than a

given threshold in cloud liquid phase the algorithm willdetermine whether to employ the reciprocal evolution If thereciprocal evolution condition is met the reciprocal evolu-tion is employed Otherwise the algorithm carries out phasetransformation In reciprocity operation the probability ofone individualrsquos reputation is calculated by 119906 119906 isin [0 1] isa uniformly random number The cost-to-benefit ratio of thealtruistic act is (1minus0618) asymp 038 and 0618 is the golden ratio119873119901represents the number of the subpopulations 119899119906119888119897119890119906119904

119894is

the nucleus of the 119894th subpopulation Consider

119899119906119888119897119890119906119904119873119901=sum119873119901minus1

119894=1

119899119906119888119897119890119906119904119894

119873119901minus 1

(9)

Generally the optima value will act as the nucleus How-ever the nucleus of the last subpopulation is the mean of theoptima values coming from the rest of the subpopulations inreciprocity operation The purpose of the reciprocity mech-anism is to ensure the diversity of the cloud particles Forexample circle represents the location of the global optimumand pentagram represents the location of the local optima inFigure 8 Figure 8(a) displays the position of nucleus beforereciprocity operator Figure 8(b) shows the individuals gen-erated by cloud generator based on Figure 8(a) Figure 8(c)shows the position of nucleus with reciprocity operator basedon Figure 8(a) Figure 8(d) shows the individuals generatedby cloud generator based on Figure 8(c) By comparingFigure 8(b) with Figure 8(d) it can be seen that the distri-bution of the individuals in Figure 8(d) is more widely thanthose in Figure 8(b)

245 Solidification Operation The algorithm realizes con-vergence operation in cloud solid phase by solidificationoperation sf is the solidification factor

119864119899119894119895=

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816

sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895

10038161003816100381610038161003816gt 10minus2

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816 otherwise

(10)

119867119890119894119895=

119864119899119894119895

100 (11)

where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873

119901represents the

number of the subpopulations 119863 represents the number ofdimensions

The pseudocode of CPEA is illustrated in Algorithms 12 3 and 4 119873

119901represents the number of the subpopulation

Childnum represents the number of individuals of everysubpopulation The search space is [minus119903 119903]

3 Experiments and Discussions

In this section the performance of the proposed CPEA istested with a comprehensive set of benchmark functionswhich include 40 different global optimization problemsand CEC2013 benchmark problems The definition of thebenchmark functions and their global optimum(s) are listedin the Appendix The benchmark functions are chosen from


2 25 3 35 4 45 5044046048

05052054056058

06

NucleusLocal optima valueGlobal optima value

x2

x1

(a)

IndividualsLocal optima valueGlobal optima value

2 25 3 35 4 45 504

045

05

055

06

065

x2

x1

(b)

25 26 27 28 29 3 31044

046

048

05

052

054

056


x2

x1

(c)


22 23 24 25 26 27 28 29 3 31 3204

045

05

055

06

x2

x1

(d)

Figure 8 Reciprocity operator

(1) if State == 0 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) if PT le 120575 then(5) Phase Transformation from gaseous to liquid(6) 119873

119901

= lceil119873119901

2rceil

(7) Childnum = Childnum times 2(8) Select the nucleus of each sub-population(9) 120575 = 12057510

(10) State = 1(11) else(12) Condensation Operation(13) for 119894 = 1119873

119875

(14) Update 119864119899119894for each sub-population according to (7)

(15) Update119867119890119894for each sub-population according to (8)

(16) endfor(17) endif(18) endif

State = 0 Cloud Gaseous Phase State = 1 Cloud Liquid Phase State = 2 Cloud Solid Phase

Algorithm 1 Evolutionary algorithm of the cloud gaseous state


(1) if State == 1 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) for 119894 = 1119873

119901

(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904

119894119895

| gt 1

(7) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|20

(8) elseif |119899119906119888119897119890119906119904119894119895

| gt 01

(9) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|3

(10) else(11) 119864

119899119894119895= |119899119906119888119897119890119906119904

119894119895

|

(12) endif(13) endfor(14) endfor(15) for 119894 = 1119873

119901


(17) endfor(18) if PT le 120575 then(19) if rand gt 119888119887 then(20) Reciprocity Operation according to (9) Reciprocity Operation(21) else(22) Phase Transformation from liquid to solid(23) 119873

119901

= lceil119873119901

2rceil


(27) State = 2(28) endif(29) endif(30) endif

Algorithm 2 Evolutionary algorithm of the cloud liquid state

(1) if State == 2 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) Solidification Operation(5) if PT le 120575 then(6) sf = sf times 2(7) 120575 = 12057510

(8) endif(9) for 119894 = 1119873

119875

(10) for 119895 = 1119863(11) Update 119864

119899119894119895according to (10)

(12) Update119867119890119894119895according to (11)

(13) endfor(14) endfor(15) endif

Algorithm 3 Evolutionary algorithm of the cloud solid state

Ali et al [25] and Rahnamayan et al [26]They are unimodalfunctions multimodal functions and rotate functions Allthe algorithms are implemented on the same machine witha T2450 24GHz CPU 15 GB memory and windows XPoperating system with Matlab R2009b

31 Parameter Study In this section the setting basis forinitial values of some parameters involved in the cloud parti-cles evolution algorithm model is described Firstly we hopethat all the cloud particles which are generated by the cloudgenerator can fall into the search space as much as possible


(1) Initialize119863 (number of dimensions)119873119901

= 5 Childnum = 30 119864119899

=23r119867119890

= 119864119899

1000(2) for 119894 = 1119873

119901

(3) for 119895 = 1119863(4) if 119895 is even(5) 119864

119909119894119895= 1199032

(6) else(7) 119864

119909119894119895= minus1199032

(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while

Algorithm 4 Cloud particles evolution algorithm CPEA

1 2 3 4 5 6 7 8 9 10965

970

975

980

985

990

995

1000

Number of run times

Valid

clou

d pa

rtic

les

1 2 3 4 5 6 7 8 9 10810

820

830

840

850

860

870

Number of run times

Valid

clou

d pa

rtic

les

En = (13)r

En = (12)r

En = (23)r

En = (34)rEn = (45)r

En = r

Figure 9 Valid cloud particles of different 119864119899

Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters

311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903

is the search radius Set 119903 = 2 1198641199091= minus23 119864

1199092= 23 119867

119890=

0001119864119899 Five hundred cloud particles are generated in every

experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of

valid cloud particles generated with different 119864119899

It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864

119899increases The average

number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the

number of valid cloud particles is relatively fewer when 119864119899is

(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864

119899is (13)119903

(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864

119899is

(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864

119899

is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in

Figure 10 represents the decision variables space

312 Definition of 119867119890 Ackley Function (multimode) and

Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867

119890 Let the search space


0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2

minus2 minus15 minus1 minus05

minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1

minus2 minus15 minus1 minus05 minus2minus2

minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3

Figure 10 Distribution of cloud particles in different 119864119899

1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es

Ackley function (D = 100)

He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs

Sphere function (D = 100)

He

(b)

Figure 11 NFES of CPEA in different119867119890

be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =

10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899

The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867

119890is equal

to 0001119864119899 For Sphere Function the number of fitness eval-

uations is less when 119867119890is equal to 0001119864

119899 Therefore 119867

119890is

equal to 0001119864119899in the cloud particles evolution algorithm


Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864

119909being equal to 119903 (12)119903

(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10

minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864

119909

is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864

119909is equal to (12)119903 or (13)119903There-

fore 119864119909is equal to (12)119903 in the cloud particles evolution

algorithm

314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically

119886 + 119887

119886=119886

119887

def= 120593 (12)

in which 120593 represents the golden ration Its value is

120593 =1 + radic5

2= 16180339887

1

120593= 120593 minus 1 = 0618 sdot sdot sdot

(13)

The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution


0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)

Figure 12 NFES of CPEA in different 119864119909

which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process

In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038

32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864

119899= (23)119903 (the search space is [minus119903 119903])

119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10

minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864

119899= 1 and 119867

119890=

1198641198996 The parameters of PSO-cf-Local and FDR-PSO come

from [31 32]

33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows

Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows

SR = successful runstotal runs

(14)

Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909

lowast is the global minimum of the function The

minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated

Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value

Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated

Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms

AR =NFES

1198861

NFES1198862

(15)

in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)

The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows

ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)

34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop

As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891

07and 11989117 However CPEA

converges slower thanCMA-ESwith respect to11989118When the


Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107

11989117

11989118

and 119891

33

function

119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA

11989107

1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260

11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850

DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360

11989118

1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248

11989133

09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300

The bold entities indicate the best results obtained by the algorithm119863 represents dimension

dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891

33 Nonetheless CPEA converges slower while the

dimension increases

35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface

As shown in Table 2 CPEA obtains better results in func-tion119891

0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains

relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864

119899is set according to

the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864

119899which can be opti-

mized in order to get better results is an important controllingparameter in the algorithm

36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30

except11989103(119863 = 2)The termination criterion is to find a value

smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface

As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891

14which successful

run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891

01 11989102 11989104 11989111 11989114

and 11989117 For other functions CPEA spends relatively long

time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864

119899given in

the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision

As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891

13and 119891

20 The reason is that the phase

transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides

the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions

As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891

25 11989126 11989127 11989128

and 11989131) This result

shows that the setting of119867119890needs to be further improved

37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891

01 11989111 11989116 11989132 and 119891

33 However

the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891

04 The solution obtained by CPEA is

worse than JADE for high dimensional problems such asfunction 119891

05and function 119891

07 The reason is that the setting

of initial population of CPEA is insufficient for high dimen-sional problems

38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables

It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067


Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101

sim11989113

119865 NFES 119863CMA-ESmeanSD

DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29

320e minus 07251e minus 07





30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110

5 k 2 699e minus 47221e minus 46






11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00




ldquoSDrdquo stands for standard deviation

079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance

39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences

between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold

Table 8 consists of unimodal problems (11986501ndash11986505) and

basic multimodal problems (11986506ndash11986520) From the results of


Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function

119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4

98e + 2 mdash mdash 18e + 472e + 2 mdash

ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3

Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121

(119863 = 2) and 11989122

(119863 = 2) for eachfunction

119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113

40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115

40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116

40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117

200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118



40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121

30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122

30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123

30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00

Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function


40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124

40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125

40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126



40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129

40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130

40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131

40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24

11989132

40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891

08 11989124 11989125 11989126 11989127 and 119891

28in JADE DE and PSO are from literature [28]


0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32

Figure 13 Convergence graph for test functions 11989101

11989104

11989116

and 11989132

(119863 = 30)

Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865

01 where PSO-FDR PSO-cf-Local

and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865

01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865

18 the cause of the outstanding

performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution

4 Conclusions

This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a

larger adjustment for 119864119899to improve the exploration ability

of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem

As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only


0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)

increase the benefit but also reduce the risk In CPEA 119864119899and

119867119890are statically set according to the priori information and

background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions

As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864

119899

and 119867119890 Through adaptive behavior these parameters can

automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems

Appendix

Test Functions

11989101

Sphere model 11989101(119909) = sum

119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102

Step function 11989102(119909) = sum

119899

119894=1

(lfloor119909119894+ 05rfloor)

2 [minus100100] min(119891

02) = 0

11989103

Beale function 11989103(119909) = [15 minus 119909

1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104

Schwefelrsquos problem 12 11989104(119909) = sum

119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0

11989105Sum of different power 119891

05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107

Griewangkrsquos function 11989107(119909) = sum

119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1

cos(119909119894radic119894) + 1 [minus600 600] min(119891

07) = 0

11989108

Hartmann function 11989108(119909) =

minussum4

119894=1

119886119894exp(minussum3

119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456


Table 8 Results for CEC2013 benchmark functions over 30 independent runs

Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA

11986501

119865meanSD

164e minus 02(+)398e minus 02

151e minus 14(asymp)576e minus 14


499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD


530e minus 14(minus)576e minus 14


115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD

202e + 01(asymp)633e minus 02


204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD

103e + 00(minus)841e minus 01

297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01

Generalmerit overcontender

3 3 7 19 3 3

ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms

119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109

Six Hump Camel back function 11989109(119909) = 4119909

2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110

Matyas function (dimension = 2) 11989110(119909) =

026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111

Zakharov function 11989111(119909) = sum

119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113

Levy function 11989113(119909) = sin2(3120587119909

1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0


11989114Schwefelrsquos problem 221 119891

14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115

Axis parallel hyperellipsoid 11989115(119909) = sum

119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0

11989116De Jongrsquos function 4 (no noise) 119891

16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117

Rastriginrsquos function 11989117(119909) = sum

119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118

Ackleyrsquos path function 11989118(119909) =

minus20 exp(minus02radicsum119899119894=1

1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0

11989119Alpine function 119891

19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120

Pathological function 11989120(119909) = sum

119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121

Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =

05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122

Camel Back-3 Three Hump Problem 11989122(119909) =

21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123

Exponential Problem 11989123(119909) =

minus exp(minus05sum119899119894=1

1199092

119894

) [minus1 1] min(11989123) = minus1

11989124

Hartmann function 2 (Dimension = 6) 11989124(119909) =

minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125

Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532

11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05

11989126Shekelrsquos Family 7119898 = 7 min(119891

26) = minus104029

11989127

Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4

4) = minus10536411989128Goldstein and Price [minus2 2] min(119891

28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129

Becker and Lago Problem 11989129(119909) = (|119909

1| minus 5)2

minus

(|1199092| minus 5)2 [minus10 le 10] min(119891

29) = 0

11989130Hosaki Problem119891

30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131

Miele and Cantrell Problem 11989131(119909) =

(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0

11989132Quartic function that is noise119891

32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133

Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)

[minus512 512] min(11989133) = 0

119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134

Multi-Gaussian Problem 11989134(119909) =

minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891

34) = minus129695 (see Table 9)

11989135

Inverted cosine wave function (Masters)11989135(119909) = minussum

119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =

minus119899 + 111989136Periodic Problem 119891

36(119909) = 1 + sin2119909

1+ sin2119909

2minus

01 exp(minus11990921

minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137

Bohachevsky 1 Problem 11989137(119909) = 119909

2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139

Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +

01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0

11989140McCormick Problem119891

40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909

1le 4 minus3 le 119909

2le

3 min(11989140) = minus19133

Conflict of Interests

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

Acknowledgments

The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)

References

[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002

[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992

[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995

[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007

[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002

[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006

[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959

[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)

[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)

[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008

[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995

[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009

[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006

[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002

[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932

[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964

[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981

[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998

[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006

[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)

[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)

[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005

[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003

[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003

[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002

[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003


[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan

[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)

[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004

[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011

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

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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


This paper which is inspired by the natural phenomenaof cloud formation presents a novel optimization algorithmto solve optimization problems called cloud particles evolu-tion algorithm (CPEA) The proposed algorithm uses phasetransformation mechanism to imitate the formation anddiversification of the cloud to solve the optimization prob-lems It is well known that the species evolution is influencedby individual fitness living environment competition andcooperation between the individuals Therefore the algo-rithm introduces competition and reciprocal mechanism inthe evolution process so as to strengthen the viability of thecloud particlesThe performance of the cloud particles evolu-tion algorithm (CPEA) is tested on the classical optimizationproblems The results are compared with other algorithmsin terms of best function value and the number of functionevaluations

The remainder of this paper is organized as follows InSection 2 the proposed CPEA and the concepts are intro-duced in detail In Section 3 the performance of the proposedalgorithm is validated by different optimization problemsThe conclusion and future research are drawn in Section 4

2 Cloud Particles Evolution Algorithm

21 Cloud Formation The idea of the proposed CPEA isinspired from cloud formation The state of the cloud parti-clesrsquo changing process [10ndash12] is shown in Figure 1

After the wet air rises it will go through a set of changesbecause of some certain reasons temperature and pressureThen the water vapor is formedWater vapor plays an impor-tant role in this process When the water vapor is saturatedit will adsorb on the cloud condensation nuclei (CNN) andthe initial cloud particles are formed Cloud particles arecondensed and sublimated continuously by absorbing watervapor around the cloud In the process of the condensationthe cloud particles are getting closer and colliding Then thelarger cloud particles are formed Ice phase is generated whenthe temperature is below 0∘C at this time The emergence ofice crystals damages the stable status of cloud phase structureIce crystals grow up quickly to be the snow crystals becauseof the condensation of the water vapor On the one handwater vapor transfers to ice crystals and ice crystals beginto grow up as supercooled water droplets evaporate On theother hand the supercooledwater dropletswill be frozen afterthe colliding to make the snow particles grow bigger If thesupercooledwater vapor which has been collided is toomanythe snow crystals will be transferred to be solid substancesThen the solid substances such as snow and hail will bedropped If the solid substances descent in the area which hasa higher temperature over 0∘C they will melt into raindropThewater in our planet will be irradiated by the sunshine andit will become the water vapor again in the air The cloud isformed and changed like this

22 Phase and Phase Transformation There are various kindsof substance in nature Substances commonly exist in threestates solid liquid and gaseous Figure 2 is a simple diagramto show howwater transforms among the three states Evapo-ration condensation andmelting are the threeways for water

Raindroplets

Cloudparticles

Snowpellets

cloudparticles

Large

Figure 1 Cloud particles transformation process

Depos

ition

Subli

mati

on

Melting

Freezing

Evap

orati

onCon

dens

ation

Figure 2 Three phases of water

transforming In order to describe the different forms of thesubstance the phase can be used as a symbol for states of sub-stances In thermodynamics phase is defined as some partswhich have the same quality [13] A phase transformation isthe transition which is from one phase to another in thermo-dynamics

Phase transformation is relevant to Gibbs free energyIn thermodynamics the Gibbs free energy [14] is a ther-modynamic potential that measures the ldquousefulrdquo or process-initiating work obtainable from a thermodynamic system ata constant temperature and pressure

23 Population Evolution Mechanism Darwin pointed outthat mutation and natural selection are the two importantfactors for the evolution which explained many natural phe-nomena successfully [15] Nowak from Harvard University


1 21

3

21

3

2

4

65

78

Cooperators


Defectors

1 r




0 2 4 6 8 100

2

4

6

8

10Gaseous phase

Cloud particle





0 2 4 6 8 100

2

4

6

8

10Liquid phase




2 3 4 5 6 7 83

4

5

6

7

8Solid phase

Hailstones

Hailstones





Liquidphase

Solidphase


Globalexploration


exploitation

Gaseousphase

Reciprocalevolution

GlobalLocal


Successfulevolution

Successfulevolution


Phasetransition

Phase transition



119909) Entropy (119864


119890) 119864119909






(119864119899) That is to say119867




Cloud particle = [1199091 1199092 119909

119898] (1)


119878 = 119904119894| Φ (119864

119899 119867119890) 119894 = 1 sdot sdot sdot 119873

119883 = 119909119894| Φ (119864


(2)



=

[[[[[[

[

Cloudparticle1

Cloudparticle2

Cloudparticle119873

]]]]]]

]

=

[[[[[[[

[

1198831

1

1198831

2

1198831

3


119898

1198832

1

1198832

2

1198832

3


119898

119883119873

1

119883119873

2

119883119873

3


119898

]]]]]]]

]

(3)



119898) can


119865119894= 119891 (119909

1

1

1199091

2

1199091

119894

) 119894 = 1 2 3 119898 (4)


119901subpopu-


119901nuclei


EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897

119894119894 = 1 2 3 119892119890119899 (5)


2


2


2

(6)




119890119894is


119864119899119894=

10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816

radic119892119890119899119890119903119886119905119894119900119899 (7)

119867119890119894=119864119899119894

cd (8)



119894is


119899119906119888119897119890119906119904119873119901=sum119873119901minus1

119894=1

119899119906119888119897119890119906119904119894

119873119901minus 1

(9)



119864119899119894119895=

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816

sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895

10038161003816100381610038161003816gt 10minus2

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816 otherwise

(10)

119867119890119894119895=

119864119899119894119895

100 (11)

where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873









2 25 3 35 4 45 5044046048

05052054056058

06


x2

x1

(a)


2 25 3 35 4 45 504

045

05

055

06

065

x2

x1

(b)

25 26 27 28 29 3 31044

046

048

05

052

054

056


x2

x1

(c)


22 23 24 25 26 27 28 29 3 31 3204

045

05

055

06

x2

x1

(d)



119901

= lceil119873119901

2rceil



119875








119901

(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904

119894119895

| gt 1

(7) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|20

(8) elseif |119899119906119888119897119890119906119904119894119895

| gt 01

(9) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|3

(10) else(11) 119864

119899119894119895= |119899119906119888119897119890119906119904

119894119895

|


119901



119901

= lceil119873119901

2rceil





(8) endif(9) for 119894 = 1119873

119875

(10) for 119895 = 1119863(11) Update 119864

119899119894119895according to (10)








= 5 Childnum = 30 119864119899

=23r119867119890

= 119864119899

1000(2) for 119894 = 1119873

119901

(3) for 119895 = 1119863(4) if 119895 is even(5) 119864

119909119894119895= 1199032

(6) else(7) 119864

119909119894119895= minus1199032



1 2 3 4 5 6 7 8 9 10965

970

975

980

985

990

995

1000

Number of run times

Valid

clou

d pa

rtic

les

1 2 3 4 5 6 7 8 9 10810

820

830

840

850

860

870

Number of run times

Valid

clou

d pa

rtic

les

En = (13)r

En = (12)r

En = (23)r

En = (34)rEn = (45)r

En = r





1199092= 23 119867

119890=









119899is (13)119903


119899is


119899







0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2




minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3


1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es


He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs


He

(b)



10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899


119890is equal



119899 Therefore 119867

119890is




119909being equal to 119903 (12)119903



119909




algorithm


119886 + 119887

119886=119886

119887

def= 120593 (12)


120593 =1 + radic5

2= 16180339887

1


(13)



0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 21

3

21

3

2

4

65

78

Cooperators


Defectors

1 r




0 2 4 6 8 100

2

4

6

8

10Gaseous phase

Cloud particle





0 2 4 6 8 100

2

4

6

8

10Liquid phase




2 3 4 5 6 7 83

4

5

6

7

8Solid phase

Hailstones

Hailstones





Liquidphase

Solidphase


Globalexploration


exploitation

Gaseousphase

Reciprocalevolution

GlobalLocal


Successfulevolution

Successfulevolution


Phasetransition

Phase transition



119909) Entropy (119864


119890) 119864119909






(119864119899) That is to say119867




Cloud particle = [1199091 1199092 119909

119898] (1)


119878 = 119904119894| Φ (119864

119899 119867119890) 119894 = 1 sdot sdot sdot 119873

119883 = 119909119894| Φ (119864


(2)



=

[[[[[[

[

Cloudparticle1

Cloudparticle2

Cloudparticle119873

]]]]]]

]

=

[[[[[[[

[

1198831

1

1198831

2

1198831

3


119898

1198832

1

1198832

2

1198832

3


119898

119883119873

1

119883119873

2

119883119873

3


119898

]]]]]]]

]

(3)



119898) can


119865119894= 119891 (119909

1

1

1199091

2

1199091

119894

) 119894 = 1 2 3 119898 (4)


119901subpopu-


119901nuclei


EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897

119894119894 = 1 2 3 119892119890119899 (5)


2


2


2

(6)




119890119894is


119864119899119894=

10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816

radic119892119890119899119890119903119886119905119894119900119899 (7)

119867119890119894=119864119899119894

cd (8)



119894is


119899119906119888119897119890119906119904119873119901=sum119873119901minus1

119894=1

119899119906119888119897119890119906119904119894

119873119901minus 1

(9)



119864119899119894119895=

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816

sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895

10038161003816100381610038161003816gt 10minus2

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816 otherwise

(10)

119867119890119894119895=

119864119899119894119895

100 (11)

where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873









2 25 3 35 4 45 5044046048

05052054056058

06


x2

x1

(a)


2 25 3 35 4 45 504

045

05

055

06

065

x2

x1

(b)

25 26 27 28 29 3 31044

046

048

05

052

054

056


x2

x1

(c)


22 23 24 25 26 27 28 29 3 31 3204

045

05

055

06

x2

x1

(d)



119901

= lceil119873119901

2rceil



119875








119901

(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904

119894119895

| gt 1

(7) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|20

(8) elseif |119899119906119888119897119890119906119904119894119895

| gt 01

(9) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|3

(10) else(11) 119864

119899119894119895= |119899119906119888119897119890119906119904

119894119895

|


119901



119901

= lceil119873119901

2rceil





(8) endif(9) for 119894 = 1119873

119875

(10) for 119895 = 1119863(11) Update 119864

119899119894119895according to (10)








= 5 Childnum = 30 119864119899

=23r119867119890

= 119864119899

1000(2) for 119894 = 1119873

119901

(3) for 119895 = 1119863(4) if 119895 is even(5) 119864

119909119894119895= 1199032

(6) else(7) 119864

119909119894119895= minus1199032



1 2 3 4 5 6 7 8 9 10965

970

975

980

985

990

995

1000

Number of run times

Valid

clou

d pa

rtic

les

1 2 3 4 5 6 7 8 9 10810

820

830

840

850

860

870

Number of run times

Valid

clou

d pa

rtic

les

En = (13)r

En = (12)r

En = (23)r

En = (34)rEn = (45)r

En = r





1199092= 23 119867

119890=









119899is (13)119903


119899is


119899







0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2




minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3


1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es


He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs


He

(b)



10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899


119890is equal



119899 Therefore 119867

119890is




119909being equal to 119903 (12)119903



119909




algorithm


119886 + 119887

119886=119886

119887

def= 120593 (12)


120593 =1 + radic5

2= 16180339887

1


(13)



0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 100

2

4

6

8

10Liquid phase




2 3 4 5 6 7 83

4

5

6

7

8Solid phase

Hailstones

Hailstones





Liquidphase

Solidphase


Globalexploration


exploitation

Gaseousphase

Reciprocalevolution

GlobalLocal


Successfulevolution

Successfulevolution


Phasetransition

Phase transition



119909) Entropy (119864


119890) 119864119909






(119864119899) That is to say119867




Cloud particle = [1199091 1199092 119909

119898] (1)


119878 = 119904119894| Φ (119864

119899 119867119890) 119894 = 1 sdot sdot sdot 119873

119883 = 119909119894| Φ (119864


(2)



=

[[[[[[

[

Cloudparticle1

Cloudparticle2

Cloudparticle119873

]]]]]]

]

=

[[[[[[[

[

1198831

1

1198831

2

1198831

3


119898

1198832

1

1198832

2

1198832

3


119898

119883119873

1

119883119873

2

119883119873

3


119898

]]]]]]]

]

(3)



119898) can


119865119894= 119891 (119909

1

1

1199091

2

1199091

119894

) 119894 = 1 2 3 119898 (4)


119901subpopu-


119901nuclei


EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897

119894119894 = 1 2 3 119892119890119899 (5)


2


2


2

(6)




119890119894is


119864119899119894=

10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816

radic119892119890119899119890119903119886119905119894119900119899 (7)

119867119890119894=119864119899119894

cd (8)



119894is


119899119906119888119897119890119906119904119873119901=sum119873119901minus1

119894=1

119899119906119888119897119890119906119904119894

119873119901minus 1

(9)



119864119899119894119895=

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816

sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895

10038161003816100381610038161003816gt 10minus2

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816 otherwise

(10)

119867119890119894119895=

119864119899119894119895

100 (11)

where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873









2 25 3 35 4 45 5044046048

05052054056058

06


x2

x1

(a)


2 25 3 35 4 45 504

045

05

055

06

065

x2

x1

(b)

25 26 27 28 29 3 31044

046

048

05

052

054

056


x2

x1

(c)


22 23 24 25 26 27 28 29 3 31 3204

045

05

055

06

x2

x1

(d)



119901

= lceil119873119901

2rceil



119875








119901

(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904

119894119895

| gt 1

(7) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|20

(8) elseif |119899119906119888119897119890119906119904119894119895

| gt 01

(9) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|3

(10) else(11) 119864

119899119894119895= |119899119906119888119897119890119906119904

119894119895

|


119901



119901

= lceil119873119901

2rceil





(8) endif(9) for 119894 = 1119873

119875

(10) for 119895 = 1119863(11) Update 119864

119899119894119895according to (10)








= 5 Childnum = 30 119864119899

=23r119867119890

= 119864119899

1000(2) for 119894 = 1119873

119901

(3) for 119895 = 1119863(4) if 119895 is even(5) 119864

119909119894119895= 1199032

(6) else(7) 119864

119909119894119895= minus1199032



1 2 3 4 5 6 7 8 9 10965

970

975

980

985

990

995

1000

Number of run times

Valid

clou

d pa

rtic

les

1 2 3 4 5 6 7 8 9 10810

820

830

840

850

860

870

Number of run times

Valid

clou

d pa

rtic

les

En = (13)r

En = (12)r

En = (23)r

En = (34)rEn = (45)r

En = r





1199092= 23 119867

119890=









119899is (13)119903


119899is


119899







0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2




minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3


1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es


He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs


He

(b)



10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899


119890is equal



119899 Therefore 119867

119890is




119909being equal to 119903 (12)119903



119909




algorithm


119886 + 119887

119886=119886

119887

def= 120593 (12)


120593 =1 + radic5

2= 16180339887

1


(13)



0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898) can


119865119894= 119891 (119909

1

1

1199091

2

1199091

119894

) 119894 = 1 2 3 119898 (4)


119901subpopu-


119901nuclei


EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897

119894119894 = 1 2 3 119892119890119899 (5)


2


2


2

(6)




119890119894is


119864119899119894=

10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816

radic119892119890119899119890119903119886119905119894119900119899 (7)

119867119890119894=119864119899119894

cd (8)



119894is


119899119906119888119897119890119906119904119873119901=sum119873119901minus1

119894=1

119899119906119888119897119890119906119904119894

119873119901minus 1

(9)



119864119899119894119895=

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816

sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895

10038161003816100381610038161003816gt 10minus2

10038161003816100381610038161003816119899119906119888119897119890119906119904

119894119895

10038161003816100381610038161003816 otherwise

(10)

119867119890119894119895=

119864119899119894119895

100 (11)

where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873









2 25 3 35 4 45 5044046048

05052054056058

06


x2

x1

(a)


2 25 3 35 4 45 504

045

05

055

06

065

x2

x1

(b)

25 26 27 28 29 3 31044

046

048

05

052

054

056


x2

x1

(c)


22 23 24 25 26 27 28 29 3 31 3204

045

05

055

06

x2

x1

(d)



119901

= lceil119873119901

2rceil



119875








119901

(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904

119894119895

| gt 1

(7) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|20

(8) elseif |119899119906119888119897119890119906119904119894119895

| gt 01

(9) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|3

(10) else(11) 119864

119899119894119895= |119899119906119888119897119890119906119904

119894119895

|


119901



119901

= lceil119873119901

2rceil





(8) endif(9) for 119894 = 1119873

119875

(10) for 119895 = 1119863(11) Update 119864

119899119894119895according to (10)








= 5 Childnum = 30 119864119899

=23r119867119890

= 119864119899

1000(2) for 119894 = 1119873

119901

(3) for 119895 = 1119863(4) if 119895 is even(5) 119864

119909119894119895= 1199032

(6) else(7) 119864

119909119894119895= minus1199032



1 2 3 4 5 6 7 8 9 10965

970

975

980

985

990

995

1000

Number of run times

Valid

clou

d pa

rtic

les

1 2 3 4 5 6 7 8 9 10810

820

830

840

850

860

870

Number of run times

Valid

clou

d pa

rtic

les

En = (13)r

En = (12)r

En = (23)r

En = (34)rEn = (45)r

En = r





1199092= 23 119867

119890=









119899is (13)119903


119899is


119899







0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2




minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3


1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es


He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs


He

(b)



10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899


119890is equal



119899 Therefore 119867

119890is




119909being equal to 119903 (12)119903



119909




algorithm


119886 + 119887

119886=119886

119887

def= 120593 (12)


120593 =1 + radic5

2= 16180339887

1


(13)



0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 25 3 35 4 45 5044046048

05052054056058

06


x2

x1

(a)


2 25 3 35 4 45 504

045

05

055

06

065

x2

x1

(b)

25 26 27 28 29 3 31044

046

048

05

052

054

056


x2

x1

(c)


22 23 24 25 26 27 28 29 3 31 3204

045

05

055

06

x2

x1

(d)



119901

= lceil119873119901

2rceil



119875








119901

(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904

119894119895

| gt 1

(7) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|20

(8) elseif |119899119906119888119897119890119906119904119894119895

| gt 01

(9) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|3

(10) else(11) 119864

119899119894119895= |119899119906119888119897119890119906119904

119894119895

|


119901



119901

= lceil119873119901

2rceil





(8) endif(9) for 119894 = 1119873

119875

(10) for 119895 = 1119863(11) Update 119864

119899119894119895according to (10)








= 5 Childnum = 30 119864119899

=23r119867119890

= 119864119899

1000(2) for 119894 = 1119873

119901

(3) for 119895 = 1119863(4) if 119895 is even(5) 119864

119909119894119895= 1199032

(6) else(7) 119864

119909119894119895= minus1199032



1 2 3 4 5 6 7 8 9 10965

970

975

980

985

990

995

1000

Number of run times

Valid

clou

d pa

rtic

les

1 2 3 4 5 6 7 8 9 10810

820

830

840

850

860

870

Number of run times

Valid

clou

d pa

rtic

les

En = (13)r

En = (12)r

En = (23)r

En = (34)rEn = (45)r

En = r





1199092= 23 119867

119890=









119899is (13)119903


119899is


119899







0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2




minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3


1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es


He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs


He

(b)



10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899


119890is equal



119899 Therefore 119867

119890is




119909being equal to 119903 (12)119903



119909




algorithm


119886 + 119887

119886=119886

119887

def= 120593 (12)


120593 =1 + radic5

2= 16180339887

1


(13)



0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901

(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904

119894119895

| gt 1

(7) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|20

(8) elseif |119899119906119888119897119890119906119904119894119895

| gt 01

(9) 119864119899119894119895= |119899119906119888119897119890119906119904

119894119895

|3

(10) else(11) 119864

119899119894119895= |119899119906119888119897119890119906119904

119894119895

|


119901



119901

= lceil119873119901

2rceil





(8) endif(9) for 119894 = 1119873

119875

(10) for 119895 = 1119863(11) Update 119864

119899119894119895according to (10)








= 5 Childnum = 30 119864119899

=23r119867119890

= 119864119899

1000(2) for 119894 = 1119873

119901

(3) for 119895 = 1119863(4) if 119895 is even(5) 119864

119909119894119895= 1199032

(6) else(7) 119864

119909119894119895= minus1199032



1 2 3 4 5 6 7 8 9 10965

970

975

980

985

990

995

1000

Number of run times

Valid

clou

d pa

rtic

les

1 2 3 4 5 6 7 8 9 10810

820

830

840

850

860

870

Number of run times

Valid

clou

d pa

rtic

les

En = (13)r

En = (12)r

En = (23)r

En = (34)rEn = (45)r

En = r





1199092= 23 119867

119890=









119899is (13)119903


119899is


119899







0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2




minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3


1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es


He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs


He

(b)



10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899


119890is equal



119899 Therefore 119867

119890is




119909being equal to 119903 (12)119903



119909




algorithm


119886 + 119887

119886=119886

119887

def= 120593 (12)


120593 =1 + radic5

2= 16180339887

1


(13)



0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= 5 Childnum = 30 119864119899

=23r119867119890

= 119864119899

1000(2) for 119894 = 1119873

119901

(3) for 119895 = 1119863(4) if 119895 is even(5) 119864

119909119894119895= 1199032

(6) else(7) 119864

119909119894119895= minus1199032



1 2 3 4 5 6 7 8 9 10965

970

975

980

985

990

995

1000

Number of run times

Valid

clou

d pa

rtic

les

1 2 3 4 5 6 7 8 9 10810

820

830

840

850

860

870

Number of run times

Valid

clou

d pa

rtic

les

En = (13)r

En = (12)r

En = (23)r

En = (34)rEn = (45)r

En = r





1199092= 23 119867

119890=









119899is (13)119903


119899is


119899







0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2




minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3


1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es


He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs


He

(b)



10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899


119890is equal



119899 Therefore 119867

119890is




119909being equal to 119903 (12)119903



119909




algorithm


119886 + 119887

119886=119886

119887

def= 120593 (12)


120593 =1 + radic5

2= 16180339887

1


(13)



0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25

012

0 05 1 15 2

012

0 05 1 15 2

005

115

2

0 1 2

01234

0 1 2

0123

0 1 2 3

012345

En = (13)r En = (12)r En = (23)r

En = (34)r En = (45)r En = r

x1 x1

x1 x1 x1

x1

x2

x2

x2

x2

x2

x2

minus1

minus2




minus15

minus15

minus1

minus1

minus05

minus05

minus3

minus1minus2minus3

minus1

minus2

minus1minus2

minus1

minus2

minus3


1 2 3 4 5 6 76000

8000

10000

12000

NFF

Es


He

(a)

1 2 3 4 5 6 73000

4000

5000

6000

7000N

FFEs


He

(b)



10minus3 1198671198901= 119864119899 1198671198902= 01119864

119899 1198671198903= 001119864

119899 1198671198904= 0001119864

119899

1198671198905= 00001119864

119899 1198671198906= 000001119864

119899 and 119867

1198907= 0000001119864

119899


119890is equal



119899 Therefore 119867

119890is




119909being equal to 119903 (12)119903



119909




algorithm


119886 + 119887

119886=119886

119887

def= 120593 (12)


120593 =1 + radic5

2= 16180339887

1


(13)



0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

2

3

4

5

6

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(a)

0

1

2

3

NFF

Es


(13)r(12)r (15)r(14)r

Ex

r

times104

(b)






119867119890= 1198641198991000 119864

119909= (12)119903 cd = 1000 120575 = 10


119899= 1 and 119867

119890=


from [31 32]




(14)







AR =NFES

1198861

NFES1198862

(15)



ARave =1

119899

119899

sum

119894=1

AR119894

SRave =1

119899

119899

sum

119894=1

SR119894

(16)







11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11989117

11989118

and 119891

33

function


11989107


11989117

09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850


11989118


11989133





dimension increases



0111989102119891041198910511989107119891091198911011989111 and119891

12while it obtains










14which successful


01 11989102 11989104 11989111 11989114



119899given in



13and 119891





25 11989126 11989127 11989128




01 11989111 11989116 11989132 and 119891

33 However











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sim11989113


DEmeanSD

PSOmeanSD

ABCmeanSD

CEBAmeanSD

CPEAmeanSD

11989101

30 k 30 587e minus 29156e minus 29






30 k 50 102e minus 28187e minus 29






11989102

10 k 30 150e minus 05366e minus 05

230e + 00178e + 00


000e + 00000e + 00

616e + 00322e + 00

000e + 00000e + 00

50 k 50 000e + 00000e + 00

165e + 00205e + 00

387e + 00265e + 00

000e + 00000e + 00

608e + 00534e + 00

000e + 00000e + 00

11989103

10 k 2 265e minus 30808e minus 31

000e + 00000e + 00





11989104

100 k 30 828e minus 27169e minus 27



136e + 00193e + 00

159e + 00793e + 00


100 k 50 229e minus 06222e minus 06


501e + 00388e + 00

119e + 02454e + 01

313e + 02394e + 02


11989105

30 k 30 819e minus 09508e minus 09






50 k 50 268e minus 08167e minus 09






11989106

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00




11989107

30 k 30 188e minus 12140e minus 12






50 k 50 000e + 00000e + 00





000e + 00000e + 00

11989108

30 k 3 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00



11989109

30 k 2 000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

000e + 00000e + 00

11989110







11989111

30 k 30 123e minus 04182e minus 04



239e + 00327e + 00

142e + 00813e + 00


50 k 50 558e minus 01155e minus 01

336e + 00155e + 00

315e + 01111e + 01

571e + 01698e + 00

205e + 01940e + 00


11989112

30 k 30 319e minus 14145e minus 14






50 k 50 141e minus 14243e minus 14






11989113

30 k 30 394e minus 11115e minus 10


134e minus 31000e + 00




50 k 50 308e minus 03380e minus 03

134e minus 31000e + 00

134e minus 31000e + 00












119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865

11989101

meanSD

11989102

meanSD

11989103

meanSD

11989104

meanSD

11989107

meanSD

11989111

meanSD

11989114

meanSD

11989116

meanSD

11989117

meanSD

JADE SR 1 1 1 1 1 098 094 1 1

NFES 28e + 460e + 2

10e + 435e + 2

57e + 369e + 2

69e + 430e + 3

33e + 326e + 2

66e + 442e + 3

40e + 421e + 3

13e + 338e + 2

13e + 518e + 3

DE SR 1 1 1 1 096 1 094 1 0

NFES 33e + 466e + 2

11e + 423e + 3

16e + 311e + 2

19e + 512e + 4

34e + 416e + 3

14e + 510e + 4

26e + 416e + 4

21e + 481e + 3 mdash

PSO SR 1 1 1 0 086 0 0 1 0

NFES 29e + 475e + 2

16e + 414e + 3

77e + 396e + 2 mdash 28e + 4


ABC SR 1 1 1 0 096 0 0 1 1

NFES 17e + 410e + 3

63e + 315e + 3

16e + 426e + 3 mdash 19e + 4

23e + 3 mdash mdash 65e + 339e + 2

39e + 415e + 4

CEBA SR 1 1 1 0 1 1 096 1 1

NFES 13e + 558e + 3

59e + 446e + 3

42e + 461e + 3 mdash 22e + 5

12e + 413e + 558e + 3

29e + 514e + 4

54e + 438e + 3

11e + 573e + 3

CPEA SR 1 1 1 1 1 1 096 1 1

NFES 43e + 351e + 2

45e + 327e + 2

57e + 330e + 3

61e + 426e + 3

77e + 334e + 2

23e + 434e + 3

34e + 416e + 3

24e + 343e + 2

10e + 413e + 3


(119863 = 2) and 11989122
























11989132


08 11989124 11989125 11989126 11989127 and 119891



0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es

CPEACMAESDE

PSOABCCEBA

1010

100

10minus10

10minus20

10minus30

f01

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es

1010

100

10minus10

10minus20

f04

CPEACMAESDE

PSOABCCEBA

0 2000 4000 6000 8000 10000NFES

Mea

n of

func

tion

valu

es 1010

10minus10

10minus30

10minus50

f16

CPEACMAESDE

PSOABCCEBA

0 6000 12000 18000 24000 30000NFES

Mea

n of

func

tion

valu

es 103

101

10minus1

10minus3

f32


11989104

11989116

and 11989132

(119863 = 30)




01119865021198650311986507

11986508 11986510 11986512 11986515 11986516 and 119865



4 Conclusions






0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5000 10000 15000 20000 25000 30000NFES

Mea

n of

func

tion

valu

es

CPEAJADEDE

PSOABCCEBA

100

10minus5

10minus10

10minus15

10minus20

10minus25

f05

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

f11

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es

104

102

100

10minus2

10minus4

10minus6

f07

CPEAJADEDE

PSOABCCEBA

0 10000 20000 30000 40000 50000NFES

Mea

n of

func

tion

valu

es 102

10minus2

10minus10

10minus14

10minus6f33


11989111

11989133

(119863 = 50) and 11989107

(119863 = 100)





119899



Appendix

Test Functions

11989101


119899

119894=1

1199092

119894

[minus100 100]min(119891

01) = 0

11989102


119899

119894=1


2 [minus100100] min(119891

02) = 0

11989103


1(1 minus 119909

2)]2

+

[225minus1199091(1minus119909

2

2

)]2

+[2625minus1199091(1minus119909

3

2

)]2 [minus45 45]

min(11989103) = 0

11989104


119899

119894=1

(sum119894

119895=1

119909119895)2

[minus65 65] min(119891

04) = 0


05(119909) = sum

119899

119894=1

|119909119894|119894+1 [minus1

1] min(11989105) = 119891(0 0) = 0

11989106

Easom function 11989106(119909) =

minus cos(1199091) cos(119909

2) exp(minus(119909

1minus 120587)2

minus (1199092minus 120587)2

) [minus100100] min(119891

06) = minus1

11989107


119899

119894=1

(1199092

119894

4000) minus

prod119899

119894=1


07) = 0

11989108


minussum4

119894=1


119895=1

119860119894119895(119909119895minus 119875119894119895)2

) [0 1] min(11989108) =

minus386278214782076 Consider119886 = [1 12 3 32]

119860 =[[[

[

3 10 30

01 10 35

3 10 30

01 10 35

]]]

]


Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table6Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989101

3087748

147716

118

3115096

1076

20290

1432

18370

1477

133381

1066

5266

11666

11989102

3041588

123124

118

054316

1077

11541

1360

6247

1666

58416

1071

2966

11402

11989103

24324

14776

1090

4904

1088

6110

1071

18050

1024

38278

1011

4350

1099

11989104

20177880

1168680

110

5231152

1077

mdash0

mdashmdash

0mdash

mdash0

mdash1106

81

1607

11989105

3025140

18328

1301

29096

1086

5910

1425

6134

1410

18037

113

91700

11479

11989106

28016

166

081

121

8248

1097

7185

1112

2540

71

032

66319

1012

5033

115

911989107

30113428

169342

096

164

149744

1076

20700

084

548

20930

096

542

209278

1054

9941

11141

11989108

33376

13580

1094

3652

1092

4090

1083

1492

1226

206859

1002

31874

1011

11989109

25352

144

681

120

6292

1085

2830

118

91077

1497

9200

1058

3856

113

911989110

23608

13288

110

93924

1092

5485

1066

10891

1033

6633

1054

1920

118

811989111

30385192

1369104

110

4498200

1077

mdash0

mdashmdash

0mdash

142564

1270

1266

21

3042

11989112

30187300

1155636

112

0244396

1076

mdash0

mdash25787

172

3157139

1119

10183

11839

11989113

30101460

17040

81

144

138308

1073

6512

11558

mdash0

mdashmdash

0mdash

4300

12360

11989114

30570290

028

72250

088

789

285108

1200

mdash0

mdashmdash

0mdash

292946

119

531112

098

1833

11989115

3096

488

153304

118

1126780

1076

18757

1514

15150

1637

103475

1093

6466

11492

11989116

21016

1996

110

21084

1094

428

1237

301

1338

3292

1031

610

116

711989117

10328844

170389

076

467

501875

096

066

mdash0

mdash7935

14144

48221

1682

3725

188

28

11989118

30169152

198296

117

2222784

1076

27250

092

621

32398

1522

1840

411

092

1340

01

1262

11989119

3041116

41

337532

112

2927230

024

044

mdash0

mdash156871

1262

148014

1278

6233

165

97

Ave

096

098

098

096

086

067

401

079

636

089

131

099

1858


Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table7Com

paris

onof

DE

ODE

RDE

PSOA

BCC

EBAand

CPEA

Theb

estresultfor

each

case

ishigh

lighted

inbo

ldface

119865119863

DE

ODE

AR

ODE

RDE

AR

RDE

PSO

AR

PSO

ABC

AR

ABC

CEBA

AR

CEBA

CPEA

AR

CPEA

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

NFE

SSR

11989120

52163

088

2024

110

72904

096

074

11960

1018

mdash0

mdash58252

1004

4216

092

051

11989121

27976

15092

115

68332

1096

8372

098

095

15944

1050

11129

1072

1025

177

811989122

23780

13396

1111

3820

1098

5755

1066

2050

118

46635

1057

855

1442

11989123

1019528

115704

112

423156

1084

9290

1210

5845

1334

17907

110

91130

11728

11989124

6mdash

0mdash

0mdash

mdash0

mdash7966

094

mdash30

181

mdashmdash

0mdash

42716

098

mdash11989125

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

4280

1mdash

mdash0

mdash5471

094

mdash11989126

4mdash

0mdash

0mdash

mdash0

mdashmdash

0mdash

8383

1mdash

mdash0

mdash5391

096

mdash11989127

44576

145

001

102

5948

1077

mdash0

mdash4850

1094

mdash0

mdash5512

092

083

11989128

24780

146

841

102

4924

1097

6770

1071

6228

1077

14248

1034

4495

110

611989129

28412

15772

114

610860

1077

mdash0

mdash1914

1440

37765

1022

4950

117

011989130

23256

13120

110

43308

1098

3275

1099

1428

1228

114261

1003

3025

110

811989131

4110

81

1232

1090

1388

1080

990

111

29944

1011

131217

1001

2662

1042

11989134

25232

15956

092

088

5768

1091

3495

115

01852

1283

9926

1053

4370

112

011989135

538532

116340

1236

46080

1084

11895

1324

25303

115

235292

110

920

951

1839

11989136

27888

14272

118

59148

1086

10910

1072

8359

1094

13529

1058

1340

1589

11989137

25284

14728

1112

5488

1097

7685

1069

1038

1509

9056

1058

1022

1517

11989138

25280

14804

1110

5540

1095

7680

1069

1598

1330

8905

1059

990

1533

11989139

1037824

124206

115

646

800

1080

mdash0

mdashmdash

0mdash

153161

1025

38420

094

098

11989140

22976

12872

110

33200

1093

2790

110

766

61

447

9032

1033

3085

1096

Ave

084

084

127

084

088

073

112

089

231

077

046

098

456




11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11986501

119865meanSD




499e + 02(+)146e + 02

000e + 00(asymp)000e + 00

000e + 00(asymp)000e + 00

000e+00000e + 00

11986502

119865meanSD

120e + 05(+)150e + 05

173e + 05(+)202e + 05

331e + 06(+)113e + 06

395e + 06(+)160e + 06

437e + 04(+)376e + 04

211e + 04(asymp)161e + 04

182e + 04112e + 04

11986503

119865meanSD

175e + 06(+)609e + 05

463e + 05(+)877e + 05

135e + 07(+)130e + 07

679e + 08(+)327e + 08

138e + 06(+)189e + 06

292e + 04(+)710e + 04

491e + 00252e + 01

11986504

119865meanSD

130e + 03(+)143e + 03

197e + 02(minus)111e + 02

116e + 04(+)319e + 03

625e + 03(+)225e + 03

154e + 02(minus)160e + 02

818e + 02(asymp)526e + 02

726e + 02326e + 02

11986505

119865meanSD




115e + 02(+)945e + 01


000e + 00(minus)000e + 00


11986506

119865meanSD

630e + 00(asymp)447e + 00

621e + 00(asymp)436e + 00

151e + 00(minus)290e + 00

572e + 01(+)194e + 01

528e + 00(asymp)492e + 00

926e + 00(+)251e + 00

621e + 00480e + 00

11986507

119865meanSD

725e minus 01(+)193e + 00

163e + 00(+)189e + 00

408e + 01(+)118e + 01

401e + 01(+)531e + 00

189e + 00(+)194e + 00

687e minus 01(+)123e + 00


11986508

119865meanSD



204e + 01(+)813e minus 02




202e + 01462e minus 02

11986509

119865meanSD


297e + 00(asymp)126e + 00

530e + 00(+)855e minus 01

683e + 00(+)571e minus 01

244e + 00(asymp)137e + 00


232e + 00129e + 00

11986510

119865meanSD



142e + 00(+)327e minus 01

624e + 01(+)140e + 01




11986511

119865meanSD



000e + 00(minus)000e + 00

456e + 01(+)589e + 00


122e + 00(minus)129e + 00

795e + 00330e + 00

11986512

119865meanSD

531e + 00(asymp)227e + 00

138e + 01(+)658e + 00

280e + 01(+)822e + 00

459e + 01(+)513e + 00

138e + 01(+)480e + 00

636e + 00(+)219e + 00

500e + 00121e + 00

11986513

119865meanSD

774e + 00(minus)459e + 00

189e + 01(asymp)614e + 00

339e + 01(+)986e + 00

468e + 01(+)805e + 00

153e + 01(asymp)607e + 00

904e + 00(minus)400e + 00

170e + 01872e + 00

11986514

119865meanSD

283e + 00(minus)336e + 00

482e + 01(minus)511e + 01


974e + 02(+)112e + 02

565e + 01(minus)834e + 01

200e + 02(minus)100e + 02

317e + 02153e + 02

11986515

119865meanSD

304e + 02(asymp)203e + 02

688e + 02(+)291e + 02

735e + 02(+)194e + 02

721e + 02(+)165e + 02

638e + 02(+)192e + 02

390e + 02(+)124e + 02

302e + 02115e + 02

11986516

119865meanSD




102e + 00(+)114e minus 01




11986517

119865meanSD

145e + 01(asymp)411e + 00

874e + 00(minus)471e + 00

855e + 00(minus)261e + 00

464e + 01(+)721e + 00

132e + 01(asymp)544e + 00

137e + 01(asymp)115e + 00

143e + 01162e + 00

11986518

119865meanSD

229e + 01(+)549e + 00

231e + 01(+)104e + 01

401e + 01(+)584e + 00

548e + 01(+)662e + 00

210e + 01(+)505e + 00

185e + 01(asymp)564e + 00

174e + 01465e + 00

11986519

119865meanSD




717e + 00(+)121e + 00




11986520

119865meanSD



340e + 00(+)240e minus 01

308e + 00(+)219e minus 01



272e + 00583e minus 01


3 3 7 19 3 3


119875 =

[[[[[

[

036890 011700 026730

046990 043870 074700

01091 08732 055470

003815 057430 088280

]]]]]

]

(A1)

11989109


2

1

minus

211199094

1

+(13)1199096

1

+11990911199092minus41199094

2

+41199094

2

[minus5 5] min(11989109) =

minus10316

11989110


026(1199092

1

+ 1199092

2

) minus 04811990911199092 [minus10 10] min(119891

10) = 0

11989111


119899

119894=1

1199092

119894

+

(sum119899

119894=1

05119894119909119894)2

+ (sum119899

119894=1

05119894119909119894)4 [minus5 10] min(119891

11) = 0

11989112


119899

119894=1

|119909119894| +

prod119899

119894=1

|119909119894| [minus10 10] min(119891

12) = 0

11989113


1) + sum

119899minus1

119894=1

(119909119894minus

1)2

times (1 + sin2(3120587119909119894+1)) + (119909

119899minus 1)2

(1 + sin2(2120587119909119899))

[minus10 10] min(11989113) = 0



14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











14(119909) = max|119909

119894| 1 le 119894 le

119899 [minus100 100] min(11989114) = 0

11989115


119899

119894=1

1198941199092

119894

[minus512 512] min(119891

15) = 0


16(119909) = sum

119899

119894=1

1198941199094

119894

[minus128 128] min(119891

16) = 0

11989117


119899

119894=1

(1199092

119894

minus

10 cos(2120587119909119894) + 10) [minus512 512] min(119891

17) = 0

11989118



1199092119894

119899) minus exp(sum119899119894=1

cos(2120587119909119894)119899) +

20 + 119890 [minus32 32] min(11989118) = 0


19(119909) = sum

119899

119894=1

|119909119894sin(119909119894) + 01119909

119894|

[minus10 10] min(11989119) = 0

11989120


119899minus1

119894

(05 +

(sin2radic(1001199092119894

+ 1199092119894+1

) minus 05)(1 + 0001(1199092

119894

minus 2119909119894119909119894+1+

1199092

119894+1

)2

)) [minus100 100] min(11989120) = 0

11989121


05+(sin2radic(11990921

+ 11990922

)minus05)(1+001(1199092

1

+1199092

2

)2

) [minus1010] min(119891

21) = 0

11989122


21199092

1

minus1051199094

1

+(16)1199096

1

+11990911199092+1199092

2

[minus5 5]min(11989122) = 0

11989123



1199092

119894

) [minus1 1] min(11989123) = minus1

11989124


minussum4

119894=1


119895=1

119861119894119895(119909119895minus 119876119894119895)2

) [0 1]

119886 = [1 12 3 32]

119861 =

[[[[[

[

10 3 17 305 17 8

005 17 10 01 8 14

3 35 17 10 17 8

17 8 005 10 01 14

]]]]]

]

119876 =

[[[[[

[

01312 01696 05569 00124 08283 05886

02329 04135 08307 03736 01004 09991

02348 01451 03522 02883 03047 06650

04047 08828 08732 05743 01091 00381

]]]]]

]

(A2)

min(11989124) = minus333539215295525

11989125


11989125(119909) = minussum

119898

119894=1

[(119909119894minus 119886119894)(119909119894minus 119886119894)119879

+ 119888119894]minus1

119886 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

4 1 8 6 3 2 5 8 6 7

4 1 8 6 7 9 5 1 2 36

4 1 8 6 3 2 3 8 6 7

4 1 8 6 7 9 3 1 2 36

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119879

(A3)

119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05

1003816100381610038161003816

119879

Table 9

119894 119886119894

119887119894

119888119894

119889119894

1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05


26) = minus104029

11989127



28) = 3

11989128(119909) = [1 + (119909

1+ 1199092+ 1)2

sdot (19 minus 141199091+ 31199092

1

minus 141199092+ 611990911199092+ 31199092

2

)]

times [30 + (21199091minus 31199092)2

sdot (18 minus 321199091+ 12119909

2

1

+ 481199092minus 36119909

11199092+ 27119909

2

2

)]

(A4)

11989129


1| minus 5)2

minus


29) = 0


30(119909) = (1minus8119909

1+71199092

1

minus(73)1199093

1

+

(14)1199094

1

)1199092

2

exp(minus1199092) 0 le 119909

1le 5 0 le 119909

2le 6

min(11989130) = minus23458

11989131


(exp(1199091) minus 1199092)4

+100(1199092minus 1199093)6

+(tan(1199093minus 1199094))4

+1199098

1

[minus1 1] min(119891

31) = 0


32(119909) = sum

119899

119894=1

1198941199094

119894

+

random[0 1) [minus128 128] min(11989132) = 0

11989133


[minus512 512] min(11989133) = 0



minussum5

119894=1

119886119894exp(minus(((119909

1minus 119887119894)2

+ (1199092minus 119888119894)2

)1198892

119894

)) [minus2 2]min(119891


11989135


119899minus1

119894=1

(exp((minus(1199092119894

+ 1199092

119894+1

+ 05119909119894119909119894+1))8) times

cos(4radic1199092119894

+ 1199092119894+1

+ 05119909119894119909119894+1)) [minus5 le 5] min(119891

35) =


36(119909) = 1 + sin2119909

1+ sin2119909

2minus


minus 1199092

2

) [minus10 10] min(11989136) = 09

11989137


2

1

+ 21199092

2

minus

03 cos(31205871199091) minus 04 cos(4120587119909

2) + 07 [minus50 50]

min(11989137) = 0

11989138


2

1

+ 21199092

2

minus

03 cos(31205871199091) cos(4120587119909

2)+03 [minus50 50]min(119891

38) = 0


11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










11989139


01119909 [minus100 100] 119909 = radicsum119899119894=1

1199092119894

min(11989139) = 0


40(119909) = sin(119909

1+1199092)+(1199091minus

1199092)2

minus(32)1199091+(52)119909

2+1 minus15 le 119909


2le

3 min(11989140) = minus19133



Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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