Review Article Evolutionary Algorithms Applied to Antennas ...Review Article Evolutionary Algorithms Applied to Antennas and Propagation: A Review of State of the Art SotiriosK.Goudos,

Review ArticleEvolutionary Algorithms Applied to Antennas and PropagationA Review of State of the Art

Sotirios K Goudos1 Christos Kalialakis2 and Raj Mittra34

1Radiocommunications Laboratory Aristotle University of Thessaloniki 54124 Thessaloniki Greece2Centre Tecnologic de Telecomunicacions de Catalunya-CTTC Castelldefels 08860 Barcelona Spain3Department of Electrical and Computer Engineering University of Central Florida Orlando FL 32816 USA4King Abdulaziz University Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Christos Kalialakis christoskalialakiscttccat

Received 16 May 2016 Accepted 11 July 2016

Academic Editor Jaume Anguera

Copyright copy 2016 Sotirios K Goudos et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A review of evolutionary algorithms (EAs) with applications to antenna and propagation problems is presented EAs have emergedas viable candidates for global optimization problems and have been attracting the attention of the research community interestedin solving real-world engineering problems as evidenced by the fact that very large number of antenna design problems havebeen addressed in the literature in recent years by using EAs In this paper our primary focus is on Genetic Algorithms (GAs)Particle Swarm Optimization (PSO) and Differential Evolution (DE) though we also briefly review other recently introducednature-inspired algorithms An overview of case examples optimized by each family of algorithms is included in the paper

1 Introduction

Several evolutionary algorithms (EAs) have emerged inthe past decade that mimic the behavior and evolution ofbiological entities inspired mainly by Darwinrsquos theory ofevolution and its natural selection mechanism The studyof evolutionary algorithms began in the 1960s Severalresearchers independently developed three mainstream evo-lutionary algorithms namely the Genetic Algorithms [1 2]Evolutionary Programming [3] and evolution strategies [4](Figure 1) EAs are widely used for the solution of single andmultiobjective optimization problems and Figure 1 depictssome of the main algorithmic families

Swarm Intelligence (SI) algorithms are also a special typeof EA The SI can be defined as the collective behaviorof decentralized and self-organized swarms SI algorithmsinclude Particle Swarm Optimization (PSO) [5] Ant ColonyOptimization [6] and Artificial Bee Colony (ABC) [7]PSO mimics the swarm behavior of birds flocking and fishschooling [5]Themost common PSO algorithms include theclassical Inertia Weight PSO (IWPSO) and the ConstrictionFactor PSO (CFPSO) [8] The PSO algorithm is easy toimplement and is computationally efficient it is typically used

only for real-valued problems An option to expand PSO fordiscrete-valued problems also exists [9] Other SI algorithmsinclude (i) Artificial Bee Colony (ABC) [7] which modelsand simulates the behaviors of honey bees foraging for foodand (ii) Ant Colony Optimization (ACO) [6 10 11] which isa population-based metaheuristic inspired by the behavior ofreal ants

Differential Evolution (DE) [12 13] is a population-basedstochastic global optimization algorithm which has beenused in several real-world engineering problems utilizingseveral variants of the DE algorithm An overview of boththe PSO and DE algorithms and hybridizations of thesealgorithms along with other soft computing tools can befound in [14]

Other evolutionary techniques applied to antenna prob-lems include the recently proposed Wind Driven Optimiza-tion (WDO) [15] biogeography-based optimization (BBO)Invasive Weed Optimization (IWO) [16ndash20] EvolutionaryProgramming (EP) [21 22] and the Covariance MatrixAdaptation Evolution Strategy (CMA-ES) [23 24]

An important theorem which is pertinent to the per-formance of optimization algorithms is the so-called ldquoNo

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 1010459 12 pageshttpdxdoiorg10115520161010459

2 International Journal of Antennas and Propagation

Evolutionary algorithms (EAs)

GeneticAlgorithms

EvolutionaryProgramming

Evolutionstrategies

DifferentialEvolution

SwarmIntelligence

CMA-ES PSO ABC ACO

Figure 1 A diagram depicting main families of evolutionary algorithms

Free Lunchrdquo (NFL) theorem which pertains to the averagebehavior of optimization algorithms over given spaces ofoptimization problems It has been shown in [25] that whenaveraged over all possible optimization problems definedover some search space 119883 no algorithm has a performanceadvantage over any other Additionally in [26] the authorsshow that it is theoretically impossible to have a best general-purpose universal optimization strategy and the only way forone strategy to be superior to the others is when it focuseson a particular class of problems It should be noted that awide variety of optimization problems arise in the antennadomain and that it is not always an easy task to find the bestoptimization algorithm for solving each of them Thereforeit is worthwhile to explore new optimization algorithms ifwe find that they can work well for the problem at handOptimization problems arising in the design and synthesis ofantennas can benefit considerably from an application of theEAs which can lead to unconventional solutions regardingposition and excitation of the antenna elements in an arrayFurthermore the elements themselves can be geometricallydesigned by using the EAs

The purpose of this paper is to briefly describe thealgorithms and present their application to antenna designproblems found in the recent literature

Section 2 reviews the Genetic Algorithms (GAs) Sec-tion 3 focuses on PSO and Section 4 is devoted to DE A briefoverview of other EA algorithms is provided in Section 5We describe the algorithms briefly and then present somestatistics regarding the use of the algorithms in the literatureAdditionally for each algorithm some representative papersare referenced in the tables Some open issues are discussedin Section 6 and conclusions are drawn in Section 7

2 Genetic Algorithms

GAs the most popular EAs are inspired by Darwinrsquos naturalselection GAs can be real or binary-coded In a binary-coded

GA each chromosome encodes a binary string [27 28]The most commonly used operators are crossover mutationand selection The selection operator chooses two parentchromosomes from the current population according to aselection strategy Most popular selection strategies includeroulette wheel and tournament selection The crossoveroperator combines the two parent chromosomes in order toproduce one new child chromosome The mutation operatoris applied with a predefined mutation probability to a newchild chromosome

A search in the Scopus database shows that there are65762 conference papers and 94510 journal papers relatedto GAs from 1977 to 2016 Figure 2 shows the numberof papers related to GAs and antennas over the last 15years Additionally the search in the same database usingthe keywords GAs and Antennas reveals a total number of2807 papers (both journal and conference) Tutorials andapplications of GAs to electromagnetics can be found in[29 30]

Table 1 lists selected papers that use GAs (mostly binary-coded) for antenna design Several papers exist in the lit-erature that apply GAs for array synthesis A binary-codedGA has been applied for linear and planar array synthesis in[31 32] among others while a GA that uses decimal operatorhas been used in [33] The problem of array thinning hasbeen addressed in the literature using binary-coded GAs in[34] while the array-failure correction problem has beenaddressed by using the real-coded GAs in [35] A simple GAhas been used for the synthesis of time-modulated arrays in[36] Different antenna types have been designed using theGAs for example wire antennas in [37ndash40] patch antennasin [41 42] and RFID tags in [43] Other GA-type algorithmssuch as a mixed integer GA have also been applied in [44]to several different antenna design problems An improvedGA for the design of linear aperiodic arrays and a binary-coded GA which uses two-point crossover (instead of theusual one point) have been used in [45] for the design of

International Journal of Antennas and Propagation 3

Table 1 Selected journal papers that use GAs for antenna design

Problem Algorithm(s) used RefThinned arrays Binary-coded GA [34]Wire antennas and Yagi-Uda antenna Binary-coded GA [37ndash40]Array synthesis Decimal operators GA [33]Array-failure correction Real-coded GA [35]

Linear and planar array synthesis Binary-coded GA and hybrid GA with simulated annealing(SA) [31 32]

Broadband patch antennas and dual-band patchantennas Binary-coded GA and parallel GA [41 42]

RFID tags Binary-coded GA [43]Time-modulated linear arrays Simple GA [36]Linear array design thinned subarrays and circularlypolarized patch antenna Mixed integer GA [44]

Linear aperiodic arrays design Improved GA [138]Low RCS slot patch antennas Binary-coded GA and two-point crossover [45]T-shaped MIMO radar antenna array chaos genetic algorithm [46]Quad-band patch antenna Binary-coded GA [47]

Pape

r num

ber

140

120

100

80

60

40

20

0

Year1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016

Journal papersConference papers

Figure 2 Papers using GAs for antenna design from 1993 to May2016

patch antennas with low RCS A GA that combines chaostheory and genetic algorithm is used for designing T-shapedMIMOradar antenna array in [46] A design procedure basedon a GA is given in [47] for optimizing a patch antennafor operation to four frequency bands GSM1800 GSM1900LTE2300 and Bluetooth

3 Particle Swarm Optimization (PSO)

In PSO the particles move in the search space where eachparticle position is updated by two optimum values The firstone is the best solution (fitness) that has been achieved so farThis value is called pbestThe other one is the global best valueobtained so far by any particle in the swarm This best value

is called 119892119887119890119904119905 After finding the pbest and 119892119887119890119904119905 the velocityupdate rule is an important factor in a PSO algorithm Themost commonly used algorithm defines that the velocity ofeach particle for every problem dimension is updated withthe following equation

119906119866+1119899119894

= 119908119906119866119899119894

+ 1198881rand1(01)

(119901119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)

+ 1198882rand2(01) (119892119887119890119904119905119866+1119899119894 minus 119909119866119899119894)

(1)

where 119906119866+1119899119894

is the 119894th particle velocity in the 119899th dimension119866 + 1 denotes the current iteration and 119866 denotes theprevious 119909

119866119899119894is the particle position in the 119899th dimension

rand1(01)

rand2(01)

are uniformly distributed random num-bers in (0 1) 119908 is a parameter known as the inertia weightand 1198881and 1198882are the learning factors

The parameter 119908 (inertia weight) is a constant between0 and 1 This parameter represents the particlersquos fly withoutany external influence The higher the value of 119908 is or thecloser it is to unity themore the particle stays unaffected frompbest and 119892119887119890119904119905 The inertia weight controls the impact of theprevious velocity a large inertia weight favors explorationwhile a small inertia weight favors exploitation The param-eter 1198881 represents the influence of the particle memory on itsbest position while the parameter 1198882 represents the influenceof the swarm best position Therefore in the Inertia WeightPSO (IWPSO) algorithm the parameters to be determined arethe swarm size (or population size) usually 100 or less thecognitive learning factor 1198881 and the social learning factor 119888

2

(usually both are set equal to 20) the inertia weight 119908 andthe maximum number of iterations It is common practice tolinearly decrease the inertia weight starting from 09 or 095to 04

Clerc [8] has suggested the use of a different velocityupdate rule which has introduced a parameter 119870 calledconstriction factor The role of the constriction factor is to


Table 2 Selected journal papers that use PSO for antenna design

Problem Algorithm(s) used RefLinear arrays design Original PSO and CLPSO [50 51 63]Profiled corrugated horn antenna design Original PSO and GA and hybrid GA-PSO [54]Phased arrays design Original PSO and GA comparison [52]Circular array design Original PSO and DE and GA comparison [53]Coplanar waveguide-fed planar monopole antenna Original PSO with MoM [56]Conformal phased arrays design Original PSO [55]Reconfigurable phase-differentiated array design Original PSO [57]Multiband and wideband patch antenna design Parallel PSOFDTD [58]Time-modulated arrays design Original PSO [59]Infinitesimal dipoles array synthesis QPSO [61]Yagi-Uda antenna CLPSO [62]Thinned planar circular arrays Modified PSO [64]Adaptive beamforming of linear antenna arrays Adaptive Mutated Boolean PSO [139]Dual-band patch antenna design Boolean PSO [140]Array design Chaotic PSO [141]Square thinned arrays Hybrid PSO and Hadamard difference sets [142]

Linear array design Feedback Particle Swarm Optimization (FPSO) with nonlinearinertia weight [65]

ensure convergence when all the particles have stopped theirmovements The velocity update rule is then given by

119906119866+1119899119894

= 119870 [119906119866119899119894

+ 1198881rand1(01)

(119901119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)

+ 1198882rand2(01)

(119892119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)]

119870 =2

10038161003816100381610038161003816100381610038162 minus 120593 minus radic1205932 minus 4120593

1003816100381610038161003816100381610038161003816

(2)

where 120593 = 1198881+ 1198882and 120593 gt 4 This PSO algorithm variant is

known as Constriction Factor PSO (CFPSO)The Scopus database shows a total number of 39673

papers from 1995 toApril 2016 for PSO related papers (includ-ing 19570 journal papers and 18355 conference papers)A refined search for antenna papers reveals 519 journalpapers and 515 conference papers from 2002 to May 2016indicating the popularity of PSO algorithm Figure 3 showsthe distribution of papers on PSO design of antennas from2002 to 2016

Table 2 lists selected PSO antenna papers Introductoryand tutorial papers that introduce the application of the PSOfor antenna design are [48 49] Additionally the problem ofsidelobe suppression of linear arrays using the PSO has beenaddressed in [50 51] A comparison of the performance of thePSO and GA algorithms as applied to the problem of phasedarrays design has been given in [52] while a comparativestudy of PSO GAs and DE for circular array design has beenreported in [53] A performance comparison of PSO GA anda hybrid GA-PSO has been provided in [54] where they havebeen applied to the problem of designing profiled corrugatedhorn antennas The application of PSO to conformal phasedarrays design has been shown in [55] A coplanar waveguide-fed planar monopole antenna for multiband operation has


Pape

r num

ber

Year2002 2004 2006 2008 2010 2012 2014 2016

0

20

40

60

80

Figure 3 Papers using PSO for antenna design from 2002 to May2016

been designed in [56] using the PSO algorithm in conjunc-tionwith theMethodofMoments (MoM)The authors in [57]use PSO for reconfigurable phase-differentiated array designA Parallelized PSO optimizer has been used in conjunctionwith the Finite Difference Time Domain (FDTD) whichhas been employed for multiband patch antenna designs in[58] and the problem of minimizing power loss in time-modulated arrays has been addressed in [59] Boundaryconditions play an important role in the application of thePSO and the performances of different boundary conditionshave been tested on a 16-element array antenna in [60] based


on mathematical benchmark functions All of the abovepapers use the original IWPSO or the CFPSO althoughseveral variants of PSO are available A new PSO variantcalled quantum PSO (QPSO) algorithm has been proposedand applied in [61] to find a set of infinitesimal dipoles whichproduces the same near and far fields as a circular dielectricresonator antenna (DRA)The interesting point about QPSOis the fact that it contains only one control parameter Thecomprehensive learning PSO (CLPSO) is applied to Yagi-Udaantenna design in [62] and unequally spaced arrays sidelobesuppression in [63] A modified PSO algorithm has beenapplied in [64] for the synthesis of thinned planar circulararrays A Feedback Particle Swarm Optimization (FPSO) isin [65] proposed for SLL minimization and null control oflinear arraysThe FPSO is based on a nonlinear inertia weightalgorithm

4 Differential Evolution (DE)

In DE the initial population evolves in each generation withthe use of three operators mutation crossover and selectionSeveral DE variants or strategies exist in the literature [13 66]that depend on the form of these operatorsThe choice of thebest DE strategy depends on the problem type [67] CommonDE strategies for the generation of trial vectors includeDErand1bin DErand-to-best2bin and DErand2binIn these strategies amutant vector V

119866+1119894for each target vector

119909119866119894

is computed by

DErand1bin

V119866+1119894 = 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) 1199031 = 1199032 = 1199033

DErand-to-best2bin

V119866+1119894

= 119909119866119894+ 119865 (119909

119866best minus 119909119866119894) + 119865 (1199091198661199031

minus 1199091198661199032

)

+ 119865 (1199091198661199033

minus 1199091198661199034

) 1199031

= 1199032

= 1199033

= 1199034

DErand2bin

V119866+1119894

= 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) + 119865 (1199091198661199034

minus 1199091198661199035

)

1199031

= 1199032

= 1199033

= 1199034

= 1199035

(3)

where 1199031 1199032 1199033 1199034 1199035are randomly chosen indices from

the population that are different from the index 119894 and119865 is a mutation control parameter After mutation thecrossover operator is applied to generate a trial vector119906119866+1119894

= (119906119866+11119894

119906119866+12119894

119906119866+1119895119894

119906119866+1119863119894

) whose coor-dinates are given by

119906119866+1119895119894

=

V119866+1119895119894

if rand119895[01)

le CR or 119895 = rn (119894)

119909119866+1119895119894

if rand119895[01)

gt CR and 119895 = rn (119894) (4)

where 119895 = 1 2 119863 rand119895[01)

is a number from auniform random distribution from the interval [0 1) rn(119894)is a randomly chosen index from (1 2 119863) and CR isthe crossover constant from the interval [0 1] DE uses a


Year2002 2004 2006 2008 2010 2012 2014 2016

Pape

r num

ber

0

5

10

15

20

25

30

35

Figure 4 Papers using DE for antenna design from 2002 to May2016

greedy selection operator which for minimization problemsis defined by

119909119866+1119894

=

119906119866+1119894

if 119891 (119906119866+1119894

) lt 119891 (119909119866119894)

119909119866119894 otherwise

(5)

where 119891(119906119866+1119894) and 119891(119909119866119894) are the fitness values of the trialand the old vector respectively The new trial vector 119906

119866+1119894

replaces the old vector 119909119866119894

only when it produces a lowerobjective-function value than the old one Otherwise the oldvector remains in the next generationThe stopping criterionfor the DE is usually the generation number or the number ofobjective-function evaluations

A search in the Scopus database reveals 38097 documentsrelated to DE (27482 journal papers and 7831 conferencepapers) A refined search for antenna related papers usingthe DE shows 221 journal papers and 152 conference papersFigure 4 shows how the papers related to DE antenna aredistributed from 2002 to May 2016

A general review paper of the use of DE in electro-magnetics has been reported in [68] and a book [69] onDE implementation in electromagnetics has been publishedTable 3 lists some representative papers for antenna designThe most commonly used DE strategy for antenna designis the DErand1bin variant The above-mentioned strat-egy has been applied among others to the problem oflinear array design in [70] synthesis of difference patternsof monopulse antennas in [71] array pattern nulling in[72] and conformal array design in [73] Several other DEstrategies have been applied to antenna problems In [74]the authors have introduced a new DDEBoR1bin strategyfor linear array synthesis while a modified DE strategy(MDES) has been used in [75] for the same problem Thestrategy DEbest1bin has been applied in [76ndash78] for time-modulated array design Self-adaptive DE algorithms havealso been applied to antenna problems including jDE [79]


Table 3 Selected journal papers that use DE for antenna design

Problem Algorithm(s) used RefLinear arrays design DErand1bin [70]Linear arrays design DDEBoR1bin [74]Linear arrays design and E-shaped patch antenna jDE [79] [80ndash82]Difference patterns of monopulse antennas DErand1bin [71]Array pattern nulling DErand1bin [72]Linear arrays design Modified DE strategy (MDES) [75]Linear arrays design Multiobjective DE [87]Array design Improved DE [143]Shaped beam synthesis CODE-EIG [86]Conformal arrays design DErand1bin [73]Horn antenna design and sparse linear arrays synthesis SaDE [83] [84 85]Subarray design Memetic GDE3 [88]Time-modulated arrays design DEbest1bin [76ndash78]Monopulse antenna with a subarray weighting Hybrid DE [144]Yagi-Uda antenna GDE3 [89]Low Radar Cross Section (RCS) slot patch antenna design DEMoM [90]

in [80ndash82] SaDE [83] in [84 85] and CODE-EIG in [86]Multiobjective DE algorithms are also another large group ofDE algorithms applied to antenna problems These includeapplications to linear array design in [87] to subarray designin [88] and to Yagi-Uda antennas [89] DE algorithmshybridized with other methods are also commonly found inthe literature for instance the DE has been used with theMethod ofMoments in [90] for the design of lowRadar CrossSection (RCS) antennas

5 Other Innovative Algorithms

Several new EAs have emerged during the last ten yearsthat are based on different evolutionary models of animalsinsects or other biological entities Artificial Bee Colony(ABC) [7] is a recently proposed SI algorithm which hasbeen applied to several real-world engineering problemsThe ABC algorithm models and simulates the honey beebehavior in food foraging In the ABC algorithm a potentialsolution to the optimization problem is represented by theposition of a food source while the food source correspondsto the quality (objective-function fitness) of the associatedsolutionThe ABC algorithm has been successfully applied toseveral problems in wireless communications [91] A numberof different variants of the ABC that improve the originalalgorithm have been proposed in [92] A search in the Scopusdatabase shows that there aremore than 3000 papers onABCof which 48 use the ABC for antenna design These includearray design [93ndash96] resonant frequency of patch antennascalculation [97] and RFID tags design [98ndash100]

Ant Colony Optimization (ACO) [6 10 11] is apopulation-based metaheuristic which was introducedby Dorigo et al [11] inspired by the behavior of real antsThe algorithm is based on the fact that ant colonies can findthe shortest path between their nest and a food source justby depositing and reacting to pheromones while they are

exploring their environment ACO is suitable for solvingcombinatorial optimization problems that are common inantennas The search in Scopus shows more than 10000papers on ACO with 169 papers dealing with the topicof antenna design The topic of linear array synthesis hasbeen presented in [101] patch antenna design in [102]sum-difference pattern synthesis in [103] and thinned arraydesign in [102] A modified touring ant colony optimizerhas been used for shaped beam synthesis [104] and forpattern nulling in [105ndash107] The authors in [108] presenta comparative study of simulated annealing (SA) GA andACO on self-structured antenna design

Biogeography-based optimization (BBO) [109] is anotherlater addition to EAs BBO is based on mathematical modelsthat describe how species migrate from one island to anotherhow new species arise and how species become extinct Theway the problem solution is found is analogous to naturersquosway of distributing speciesThe search in the Scopus databaseyielded 654 papers that refer to BBO from 2007 to May 2016with 40 papers that use BBO for antenna design from 2009till today BBO has been applied to Yagi-Uda design [110]and array synthesis [111ndash117] Additionally a hybrid DEBBOalgorithm has been used for the design of a reconfigurableantenna array with discrete phase shifters in [118]

Evolutionary Programming (EP) was originally proposedby Fogel in [3] EP is based on the idea of representingindividuals phenotypically as finite state machines capableof responding to environmental stimuli The representationsused in EP are problem-dependent The most commonrepresentation used is a fixed-length real-valued vector InEP the vectors evolve but do not exchange informationbetween them There are no crossover operators but onlymutation operators EP has been applied to several problemsin electromagnetics [21 119 120] Among others EP has beenapplied to patch antenna design [121] and to widebandparasitic dipole arrays [122]


Hansen and Ostermeier [123] introduced the CovarianceMatrix Adaptation Evolution Strategy (CMA-ES) CMA-ESis a second-order approach estimating a positive covariancematrix within an iterative procedure More details about theCMA-ES performance algorithm can be found in [124 125]In [23 24] the authors present an approach for mixed-parameter optimization based on CMA-ES which is success-fully applied in several design problems in electromagneticsThis approach is based on the concepts presented in [22]for EP The CMA-ES algorithm has been recently applied todesign problems in antennas and electromagnetics in general[126ndash129]

51 Other Artificial Intelligence Methods Other artificialintelligencemethods and techniques includeArtificialNeuralNetwork (ANN) architectures [130] which are a family ofmodels inspired by biological neural networks ANNs areused to estimate or approximate functions that can dependon a large number of inputs and are generally unknown Theapplications of ANNs to electromagnetics are a popular topicin the literature [131]

Deep learning is a type ofmachine learning based on a setof algorithms that attempt tomodel high-level abstractions indata by using a deep graph with multiple processing layerscomposed of multiple linear and nonlinear transformations[132ndash135] These methods have dramatically improved thestate of the art in speech recognition visual object recogni-tion object detection and many other domains such as drugdiscovery and genomics [132]

6 Discussion Open Issues

The choice of the best algorithm for every problem requiresthe consideration of its specific characteristics Once thealgorithm is chosen the key issue becomes the selection ofthe algorithm control parameters which inmost cases can bealso problem-dependent A practical initial approach is to usethe control parameters for these algorithms that commonlyperform well regardless of the characteristics of the problemto be solved

For real-codedGAs typical values are 09 for the crossoverprobability and 1119873 for the mutation probability where 119873is the problem dimension For the binary-coded GA typicalvalues for crossover and mutation probabilities are [10 08]and [001 01] respectively

In the PSO algorithms 1198881 and 1198882 are set equal to 205 ForCFPSO these values result in 119870 = 07298 For IWPSO it iscommon practice to linearly decrease the inertia weight start-ing from 095 to 04 The velocity is updated asynchronouslywhich means that the global best position is updated at themoment it is found

For the DE Storn and Price [13] have suggested choosingthe differential evolution control parameters 119865 and CR fromthe intervals [05 1] and [08 1] respectively and setting119873119875 = 10119863 The correct selection of these control parametervalues is frequently a problem-dependent task Multiplealgorithm runs are often required for fine-tuning the controlparameters There are several DE algorithms in the literature

that self-adapt these control parameters Another question isthe selection of the appropriate strategy for the generation oftrial vectors which requires additional computational timeusing a trial-and-error search procedure Therefore it is notalways an easy task to fine-tune the control parameters andstrategy Since finding the suitable control parameter valuesand strategy in such a way is often very time-consumingthere has been an increasing trend among researchers indesigning new adaptive and self-adaptive DE variants A DEstrategy (jDE) that self-adapts the control parameters hasbeen introduced in [79] This algorithm has been appliedsuccessfully to a microwave absorber design problem [136]and linear array synthesis [82] SaDE a DE algorithm thatself-adapts both control parameters and strategy based onlearning experiences from previous generations is presentedin [83]

The research domain of evolutionary algorithms is grow-ing rapidly A current and growing research trend in evolu-tionary algorithms is their hybridization with local optimiz-ers These algorithms are called Memetic Algorithms (MAs)[90] that are inspired by Dawkinsrsquo notion of meme Theadvantage of such an approach is that the use of a localsearch optimizer ensures that specific regions of the searchspace can be explored using fewer evaluations and goodquality solutions can be generated early during the searchFurthermore the global search algorithm generates goodinitial solutions for the local search MAs can be highlyefficient due to this combination of global exploration andlocal exploitation

An interesting idea has been presented in [137] wherethe authors conceptually present the equivalences of variouspopular EAs like GAs PSO DE and BBO Their basic con-clusion is that the conceptual equivalence of the algorithms issupported by the fact that modifications in algorithms resultin very different performance levels

Finally another concern that is pertinent to all of theabove algorithms is the definition of the stopping criterionUsually this is the iteration number or the number ofobjective-function evaluations Additionally and in orderto avoid stagnation another criterion could be set for thealgorithm to stop after a number of generations when theobjective function does not further improve

7 Conclusion

A brief survey of different evolutionary algorithms and theirapplication to different problems in antennas and propaga-tion have been presented in this review paper It is to benoted that among the evolutionary algorithms used in theliterature the GA and SI algorithms are among those mostcommonly utilized

The bibliography statistics show that GAs PSO and DEare among themost popular algorithms for antenna design Itmust be pointed out that several variants of these algorithmshave also been employed along with other nature-inspiredalgorithms that have emerged Most notably ABC ACOBBO EP and CMA ES have been applied to several antennadesign problems The body of literature on EAs to antenna


design is by now quite extensive and it continues to growfast with more innovative algorithms The above presentedalgorithms have been proven effective in solving specificantenna design problems although the NFL theorem assuresthat a best global optimizer does not exist the search for newalgorithms and their application to antenna design problemsis an ongoing research process which is likely to continueunabated for some time to come

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work of Christos Kalialakis was supported by the EUHorizon 2020 Research and Innovation Programme underthe Marie Sklodowska-Curie Grant Agreement no 654734

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[2] J H Holland Adaptation in Natural and Artificial Systems TheUniversity of Michigan Press Ann Arbor Mich USA 1975

[3] D B Fogel Evolutionary Computation Toward a New Philos-ophy of Machine Intelligence IEEE Press Piscataway NJ USA1995

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

[5] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Piscataway NJ USA 1995

[6] M Dorigo and T Stutzle Ant Colony Optimization The MITPress Cambridge Mass USA 2004

[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] M Clerc ldquoThe swarm and the queen towards a deterministicand adaptive particle swarm optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo99) pp 1951ndash1957IEEE Washington DC USA July 1999

[9] J Kennedy and R C Eberhart ldquoDiscrete binary version ofthe particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and Cybernetics pp4104ndash4108 Orlando Fla USA October 1997

[10] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[11] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no 1pp 29ndash41 1996

[12] R Storn and K Price ldquoDifferential EvolutionmdashA Simple andefficient adaptive scheme for global optimization over continu-ous spacesrdquo 1995 httpciteseeristpsueduarticlestorn95dif-ferentialhtml

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

[14] S Das A Abraham and A Konar ldquoParticle swarm optimiza-tion and differential evolution algorithms technical analysisapplications and hybridization perspectivesrdquo in Advances ofComputational Intelligence in Industrial Systems Y L Liu A SSun E P L Lim H T L Loh andW F L Lu Eds vol 116 pp1ndash38 Springer Berlin Germany 2008

[15] Z Bayraktar M Komurcu J A Bossard and D H WernerldquoThe wind driven optimization technique and its application inelectromagneticsrdquo IEEE Transactions on Antennas and Propa-gation vol 61 no 5 pp 2745ndash2757 2013

[16] Z D Zaharis C Skeberis T D Xenos P I Lazaridis andJ Cosmas ldquoDesign of a novel antenna array beamformerusing neural networks trained by modified adaptive dispersioninvasive weed optimization based datardquo IEEE Transactions onBroadcasting vol 59 no 3 pp 455ndash460 2013

[17] Z D Zaharis P I Lazaridis J Cosmas C Skeberis andT D Xenos ldquoSynthesis of a near-optimal high-gain antennaarray with main lobe tilting and null filling using taguchiinitialized invasive weed optimizationrdquo IEEE Transactions onBroadcasting vol 60 no 1 pp 120ndash127 2014

[18] Y-Y Bai S Xiao C Liu and B-Z Wang ldquoA hybrid IWOPSOalgorithm for pattern synthesis of conformal phased arraysrdquoIEEE Transactions on Antennas and Propagation vol 61 no 4pp 2328ndash2332 2013

[19] G G Roy S Das P Chakraborty and P N Suganthan ldquoDesignof non-uniform circular antenna arrays using a modifiedinvasive weed optimization algorithmrdquo IEEE Transactions onAntennas and Propagation vol 59 no 1 pp 110ndash118 2011

[20] S Karimkashi andAAKishk ldquoInvasiveweed optimization andits features in electromagneticsrdquo IEEE Transactions onAntennasand Propagation vol 58 no 4 pp 1269ndash1278 2010

[21] A Hoorfar ldquoEvolutionary programming in electromagneticoptimization a reviewrdquo IEEE Transactions on Antennas andPropagation vol 55 no 3 pp 523ndash537 2007

[22] A Hoorfar J Zhu and S Nelatury ldquoElectromagnetic opti-mization using a mixed-parameter self-adaptive evolutionaryalgorithmrdquo Microwave and Optical Technology Letters vol 39no 4 pp 267ndash271 2003

[23] E Boudaher and A Hoorfar ldquoElectromagnetic design opti-mization using mixed-parameter and multiobjective CMA-ESrdquoin Proceedings of the IEEE Antennas and Propagation SocietyInternational Symposium (APSURSI rsquo13) pp 406ndash407 IEEEOrlando Fla USA July 2013

[24] E BouDaher and A Hoorfar ldquoElectromagnetic optimizationusing mixed-parameter and multiobjective covariance matrixadaptation evolution strategyrdquo IEEE Transactions on Antennasand Propagation vol 63 no 4 pp 1712ndash1724 2015

[25] D HWolpert andWGMacready ldquoNo free lunch theorems foroptimizationrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 67ndash82 1997

[26] Y C Ho and D L Pepyne ldquoSimple explanation of the no-free-lunch theorem and its implicationsrdquo Journal of OptimizationTheory and Applications vol 115 no 3 pp 549ndash570 2002

[27] R L Haupt and S E Haupt Practical Genetic Algorithms 1998[28] R L Haupt and D H Werner Genetic Algorithms in Electro-

magnetics Wiley-Interscience New York NY USA 2007


[29] R L Haupt ldquoAn introduction to genetic algorithms for electro-magneticsrdquo IEEE Antennas and Propagation Magazine vol 37no 2 pp 7ndash15 1995

[30] J M Johnson and Y Rahmat-Samii ldquoGenetic algorithms inengineering electromagneticsrdquo IEEE Antennas and PropagationMagazine vol 39 no 4 pp 7ndash21 1997

[31] F J Ares-Pena J A Rodriguez-Gonzalez E Villanueva-Lopezand S R Rengarajan ldquoGenetic algorithms in the design andoptimization of antenna array patternsrdquo IEEE Transactions onAntennas and Propagation vol 47 no 3 pp 506ndash510 1999

[32] D Marcano and F Duran ldquoSynthesis of antenna arrays usinggenetic algorithmsrdquo IEEE Antennas and Propagation Magazinevol 42 no 3 pp 12ndash20 2000

[33] K-K Yan and Y Lu ldquoSidelobe reduction in array-pattern syn-thesis using genetic algorithmrdquo IEEE Transactions on Antennasand Propagation vol 45 no 7 pp 1117ndash1122 1997

[34] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[35] B-K Yeo and Y Lu ldquoArray failure correction with a geneticalgorithmrdquo IEEE Transactions on Antennas and Propagationvol 47 no 5 pp 823ndash828 1999

[36] S Yang Y B Gan A Qing and P K Tan ldquoDesign of a uniformamplitude time modulated linear array with optimized timesequencesrdquo IEEE Transactions on Antennas and Propagationvol 53 no 7 pp 2337ndash2339 2005

[37] A Boag A Boag E Michielssen and R Mittra ldquoDesign ofelectrically loaded wire antennas using genetic algorithmsrdquoIEEE Transactions on Antennas and Propagation vol 44 no 5pp 687ndash695 1996

[38] E E Altshuler and D S Linden ldquoWire-antenna designs usinggenetic algorithmsrdquo IEEE Antennas and Propagation Magazinevol 39 no 2 pp 33ndash43 1997

[39] E A Jones and W T Joines ldquoDesign of yagi-uda antennasusing genetic algorithmsrdquo IEEE Transactions on Antennas andPropagation vol 45 no 9 pp 1386ndash1392 1997

[40] E E Altshuler ldquoElectrically small self-resonant wire antennasoptimized using a genetic algorithmrdquo IEEE Transactions onAntennas and Propagation vol 50 no 3 pp 297ndash300 2002

[41] J Michael Johnson ldquoGenetic algorithms and method ofmoments (GAMOM) for the design of integrated antennasrdquoIEEE Transactions on Antennas and Propagation vol 47 no 10pp 1606ndash1614 1999

[42] F J Villegas T Cwik Y Rahmat-Samii and M Manteghi ldquoAparallel electromagnetic genetic-algorithmoptimization (EGO)application for patch antenna designrdquo IEEE Transactions onAntennas and Propagation vol 52 no 9 pp 2424ndash2435 2004

[43] G Marrocco ldquoGain-optimized self-resonant meander lineantennas for RFID applicationsrdquo IEEE Antennas and WirelessPropagation Letters vol 2 pp 302ndash305 2003

[44] R L Haupt ldquoAntenna design with a mixed integer geneticalgorithmrdquo IEEE Transactions on Antennas and Propagationvol 55 no 3 pp 577ndash582 2007

[45] X ZhuW Shao J-L Li andYDong ldquoDesign and optimizationof low RCS patch antennas based on a genetic algorithmrdquoProgress in Electromagnetics Research vol 122 pp 327ndash339 2012

[46] X Fu X Chen Q Hou Z Wang and Y Yin ldquoAn improvedchaos genetic algorithm for T-shaped MIMO radar antennaarray optimizationrdquo International Journal of Antennas andPropagation vol 2014 Article ID 631820 6 pages 2014

[47] J M J W Jayasinghe J Anguera D N Uduwawala andA Andujar ldquoNonuniform overlapping method in designingmicrostrip patch antennas using genetic algorithm optimiza-tionrdquo International Journal of Antennas and Propagation vol2015 Article ID 805820 8 pages 2015

[48] N Jin and Y Rahmat-Samii ldquoAdvances in particle swarmoptimization for antenna designs real-number binary single-objective and multiobjective implementationsrdquo IEEE Transac-tions on Antennas and Propagation vol 55 no 3 pp 556ndash5672007

[49] J Robinson andY Rahmat-Samii ldquoParticle swarmoptimizationin electromagneticsrdquo IEEE Transactions on Antennas and Prop-agation vol 52 no 2 pp 397ndash407 2004

[50] M M Khodier and C G Christodoulou ldquoLinear array geom-etry synthesis with minimum sidelobe level and null controlusing particle swarm optimizationrdquo IEEE Transactions onAntennas and Propagation vol 53 no 8 pp 2674ndash2679 2005

[51] P J Bevelacqua andC A Balanis ldquoMinimum sidelobe levels forlinear arraysrdquo IEEE Transactions on Antennas and Propagationvol 55 no 12 pp 3442ndash3449 2007

[52] D W Boeringer and D H Werner ldquoParticle swarm optimiza-tion versus genetic algorithms for phased array synthesisrdquo IEEETransactions on Antennas and Propagation vol 52 no 3 pp771ndash779 2004

[53] M A Panduro C A Brizuela L I Balderas and D A AcostaldquoA comparison of genetic algorithms particle swarm optimiza-tion and the differential evolution method for the design ofscannable circular antenna arraysrdquo Progress in ElectromagneticsResearch B vol 13 pp 171ndash186 2009

[54] J Robinson S Sinton and Y Rahmat-Samii ldquoParticle swarmgenetic algorithm and their hybrids optimization of a profiledcorrugated horn antennardquo in Proceedings of the IEEE Antennasand Propagation Society International Symposium pp 314ndash317San Antonio Tex USA June 2002

[55] D W Boeringer and D H Werner ldquoEfficiency-constrainedparticle swarm optimization of a modified Bernstein polyno-mial for conformal array excitation amplitude synthesisrdquo IEEETransactions on Antennas and Propagation vol 53 no 8 pp2662ndash2673 2005

[56] W-C Liu ldquoDesign of amultiband CPW-fedMonopole antennausing a particle swarm optimization approachrdquo IEEE Transac-tions on Antennas and Propagation vol 53 no 10 pp 3273ndash3279 2005

[57] D Gies and Y Rahmat-Samii ldquoParticle swarm optimization forreconfigurable phase-differentiated array designrdquo Microwaveand Optical Technology Letters vol 38 no 3 pp 168ndash175 2003

[58] N Jin and Y Rahmat-Samii ldquoParallel particle swarm optimiza-tion and finite-difference time-domain (PSOFDTD) algorithmfor multiband and wide-band patch antenna designsrdquo IEEETransactions on Antennas and Propagation vol 53 no 11 pp3459ndash3468 2005

[59] L Poli P Rocca L Manica and A Massa ldquoHandling sidebandradiations in time-modulated arrays through particle swarmoptimizationrdquo IEEE Transactions on Antennas and Propagationvol 58 no 4 pp 1408ndash1411 2010

[60] S Xu and Y Rahmat-Samii ldquoBoundary conditions in particleswarm optimization revisitedrdquo IEEE Transactions on Antennasand Propagation vol 55 no 3 pp 760ndash765 2007

[61] S M Mikki and A A Kishk ldquoQuantum particle swarm opti-mization for electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 54 no 10 pp 2764ndash2775 2006


[62] S Baskar A Alphones PM Suganthan and J J Liang ldquoDesignof Yagi-Uda antennas using comprehensive learning particleswarm optimisationrdquo IEE ProceedingsmdashMicrowaves Antennasand Propagation vol 152 no 5 pp 340ndash346 2005

[63] S K Goudos V Moysiadou T Samaras K Siakavara and JN Sahalos ldquoApplication of a comprehensive learning particleswarm optimizer to unequally spaced linear array synthesiswith sidelobe level suppression andnull controlrdquo IEEEAntennasand Wireless Propagation Letters vol 9 pp 125ndash129 2010

[64] N Pathak G K Mahanti S K Singh J K Mishra andA Chakraborty ldquoSynthesis of thinned planar circular arrayantennas usingmodified particle swarm optimizationrdquo Progressin Electromagnetics Research Letters vol 12 pp 87ndash97 2009

[65] C Han and L Wang ldquoArray pattern synthesis using particleswarmoptimizationwith dynamic inertia weightrdquo InternationalJournal of Antennas and Propagation vol 2016 Article ID1829458 7 pages 2016

[66] R Storn ldquoDifferential evolution researchmdashtrends and openquestionsrdquo Studies in Computational Intelligence vol 143 pp 1ndash31 2008

[67] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference (GECCO rsquo06) pp 485ndash492 Seattle Wash USA July 2006

[68] P Rocca G Oliveri and A Massa ldquoDifferential evolution asapplied to electromagneticsrdquo IEEE Antennas and PropagationMagazine vol 53 no 1 pp 38ndash49 2011

[69] A Qing Differential Evolution Fundamentals and Applicationsin Electrical Engineering John Wiley amp Sons New York NYUSA 2009

[70] D G Kurup M Himdi and A Rydberg ldquoSynthesis of uniformamplitude unequally spaced antenna arrays using the differen-tial evolution algorithmrdquo IEEE Transactions on Antennas andPropagation vol 51 no 9 pp 2210ndash2217 2003

[71] S Caorsi AMassaM Pastorino and A Randazzo ldquoOptimiza-tion of the difference patterns for monopulse antennas by ahybrid realinteger-coded differential evolution methodrdquo IEEETransactions on Antennas and Propagation vol 53 no 1 pp372ndash376 2005

[72] S Yang Y B Gan and A Qing ldquoAntenna-array pattern nullingusing a differential evolution algorithmrdquo International Journalof RF and Microwave Computer-Aided Engineering vol 14 no1 pp 57ndash63 2004

[73] J-L Guo and J-Y Li ldquoPattern synthesis of conformal arrayantenna in the presence of platform using differential evolutionalgorithmrdquo IEEE Transactions on Antennas and Propagationvol 57 no 9 pp 2615ndash2621 2009

[74] C Lin A Qing and Q Feng ldquoSynthesis of unequally spacedantenna arrays by using differential evolutionrdquo IEEE Transac-tions on Antennas and Propagation vol 58 no 8 pp 2553ndash25612010

[75] Y Chen S Yang and Z Nie ldquoThe application of a modifieddifferential evolution strategy to some array pattern synthesisproblemsrdquo IEEETransactions onAntennas and Propagation vol56 no 7 pp 1919ndash1927 2008

[76] S Yang Y B Gan and A Qing ldquoSideband suppression in time-modulated linear arrays by the differential evolution algorithmrdquoIEEE Antennas and Wireless Propagation Letters vol 1 pp 173ndash175 2002

[77] G Li S Yang Y Chen and Z Nie ldquoA novel electronic beamsteering technique in time modulated antenna arraysrdquo Progressin Electromagnetics Research vol 97 pp 391ndash405 2009

[78] S Yang Y B Gan and P K Tan ldquoA new technique forpower-pattern synthesis in time-modulated linear arraysrdquo IEEEAntennas and Wireless Propagation Letters vol 2 pp 285ndash2872003

[79] 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

[80] S K Goudos A A Nanos T Samaras K Siakavara E Vafiadisand J N Sahalos ldquoUnequally spaced arrays synthesis usingself-adaptive differential evolutionrdquo in Proceedings of the 5thEuropean Conference on Antennas and Propagation (EUCAPrsquo11) pp 983ndash986 April 2011

[81] S K Goudos K Siakavara T Samaras E E Vafiadis andJ N Sahalos ldquoSelf-adaptive differential evolution applied toreal-valued antenna and microwave design problemsrdquo IEEETransactions on Antennas and Propagation vol 59 no 4 pp1286ndash1298 2011

[82] N Dib S K Goudos and H Muhsen ldquoApplication of Taguchirsquosoptimization method and self-adaptive differential evolution tothe synthesis of linear antenna arraysrdquo Progress in Electromag-netics Research vol 102 pp 159ndash180 2010

[83] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[84] S K Goudos K Siakavara T Samaras E E Vafiadis and J NSahalos ldquoSparse linear array synthesis withmultiple constraintsusing differential evolution with strategy adaptationrdquo IEEEAntennas andWireless Propagation Letters vol 10 pp 670ndash6732011

[85] D Pappas S K Goudos K B Baltzis and K SiakavaraldquoDesign of optimum gain pyramidal horn using self-adaptivedifferential evolution algorithmsrdquo International Journal of RFand Microwave Computer-Aided Engineering vol 21 no 1 pp59ndash66 2011

[86] S K Goudos ldquoShaped beam pattern synthesis of antennaarrays using composite differential evolution with eigenvector-based crossover operatorrdquo International Journal of Antennas andPropagation vol 2015 Article ID 295012 10 pages 2015

[87] S Pal B Y Qu S Das and P N Suganthan ldquoOptimal synthesisof linear antenna arrays with multi-objective differential evolu-tionrdquo Progress in Electromagnetics Research B vol 21 pp 87ndash1112010

[88] S K Goudos K A Gotsis K Siakavara E E Vafiadis and JN Sahalos ldquoA multi-objective approach to subarrayed linearantenna arrays design based onmemetic differential evolutionrdquoIEEE Transactions on Antennas and Propagation vol 61 no 6pp 3042ndash3052 2013

[89] S K Goudos K Siakavara E E Vafiadis and J N SahalosldquoPareto optimal yagi-uda antenna design using multi-objectivedifferential evolutionrdquo Progress in Electromagnetics Researchvol 105 pp 231ndash251 2010

[90] WWang S Gong XWang YGuan andW Jiang ldquoDifferentialevolution algorithm and method of moments for the designof low-RCS antennardquo IEEE Antennas and Wireless PropagationLetters vol 9 pp 295ndash298 2010


[91] Y Wang W Chen and C Tellambura ldquoA PAPR reductionmethod based on artificial bee colony algorithm for OFDMsignalsrdquo IEEE Transactions on Wireless Communications vol 9no 10 pp 2994ndash2999 2010

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

[93] B Basu and G K Mahanti ldquoFire fly and artificial bees colonyalgo-rithm for synthesis of scanned and broad-side linear arrayantennardquo Progress In Electromagnetics Research B vol 32 pp169ndash190 2011

[94] X Li and M Yin ldquoHybrid differential evolution with artificialbee colony and its application for design of a reconfigurableantenna array with discrete phase shifterdquo IET MicrowavesAntennas and Propagation vol 6 no 14 pp 1573ndash1582 2012

[95] J Yang W-T Li X-W Shi L Xin and J-F Yu ldquoA hybridABC-DE algorithm and its application for time-modulatedarrays pattern synthesisrdquo IEEE Transactions on Antennas andPropagation vol 61 no 11 pp 5485ndash5495 2013

[96] M A Zaman M Gaffar M M Alam S A Mamun and M AMatin ldquoSynthesis of antenna arrays using artificial bee colonyoptimization algorithmrdquo International Journal of Microwaveand Optical Technology vol 6 no 4 pp 234ndash241 2011

[97] A Akdagli and A Toktas ldquoA novel expression in calculatingresonant frequency of H-shaped compact microstrip antennasobtained by using artificial bee colony algorithmrdquo Journal ofElectromagnetic Waves and Applications vol 24 no 14-15 pp2049ndash2061 2010

[98] S K Goudos K Siakavara and J N Sahalos ldquoNovel spiralantenna design using artificial bee colony optimization forUHFRFID applicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 13 pp 528ndash531 2014

[99] S K Goudos K Siakavara and J N Sahalos ldquoDesign ofload-ended spiral antennas for RFID UHF passive tags usingimproved artificial bee colony algorithmrdquo AEUmdashInternationalJournal of Electronics and Communications vol 69 no 1 pp206ndash214 2015

[100] S K Goudos K Siakavara ATheopoulos E E Vafiadis and JN Sahalos ldquoApplication of Gbest-guided artificial bee colonyalgorithm to passive UHF RFID tag designrdquo InternationalJournal of Microwave andWireless Technologies vol 8 no 3 pp537ndash545 2016

[101] E Rajo-lglesias and O Quevedo-Teruel ldquoLinear array synthe-sis using an ant-colony-optimization-based algorithmrdquo IEEEAntennas and Propagation Magazine vol 49 no 2 pp 70ndash792007

[102] O Quevedo-Teruel and E Rajo-Iglesias ldquoAnt colony optimiza-tion in thinned array synthesis with minimum sidelobe levelrdquoIEEE Antennas and Wireless Propagation Letters vol 5 no 1pp 349ndash352 2006

[103] P Rocca L Manica and A Massa ldquoAn improved excitationmatching method based on an ant colony optimization forsuboptimal-free clustering in sum-difference compromise syn-thesisrdquo IEEE Transactions on Antennas and Propagation vol 57no 8 pp 2297ndash2306 2009

[104] A A Akdagli K Guney and D Karaboga ldquoTouring ant colonyoptimization algorithm for shaped-beam pattern synthesis oflinear antennardquo Electromagnetics vol 26 no 8 pp 615ndash6282006

[105] A Akdagli K Guney and D Karaboga ldquoPattern nulling oflinear antenna arrays by controlling only the element positions

with the use of improved touring ant colony optimizationalgorithmrdquo Journal of Electromagnetic Waves and Applicationsvol 16 no 10 pp 1423ndash1441 2002

[106] D Karaboga K Guney and A Akdagli ldquoAntenna array patternnulling by controlling both amplitude and phase using modi-fied touring ant colony optimization algorithmrdquo InternationalJournal of Electronics vol 91 no 4 pp 241ndash251 2004

[107] N Karaboga K Guney and A Akdagli ldquoNull steering of linearantenna arrays with use of modified touring ant colony opti-mization algorithmrdquo International Journal of RF andMicrowaveComputer-Aided Engineering vol 12 no 4 pp 375ndash383 2002

[108] C M Coleman E J Rothwell and J E Ross ldquoInvestigationof simulated annealing ant-colony optimization and geneticalgorithms for self-structuring antennasrdquo IEEE Transactions onAntennas and Propagation vol 52 no 4 pp 1007ndash1014 2004

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

[110] U Singh H Kumar and T S Kamal ldquoDesign of Yagi-Udaantenna using biogeography based optimizationrdquo IEEE Trans-actions on Antennas and Propagation vol 58 no 10 pp 3375ndash3379 2010

[111] U Singh H Kumar and T S Kamal ldquoLinear array synthesisusing biogeography based optimizationrdquo Progress in Electro-magnetics Research M vol 11 pp 25ndash36 2010

[112] A Sharaqa and N Dib ldquoDesign of linear and elliptical antennaarrays using biogeography based optimizationrdquoArabian Journalfor Science and Engineering vol 39 no 4 pp 2929ndash2939 2014

[113] N Dib and A Sharaqa ldquoDesign of non-uniform concentriccircular antenna arrays with optimal sidelobe level reductionusing biogeography-based optimizationrdquo International JournalofMicrowave andWireless Technologies vol 7 no 2 pp 161ndash1662015

[114] A Sharaqa and N Dib ldquoPosition-only side lobe reduction ofa uniformly excited elliptical antenna array using evolutionaryalgorithmsrdquo IET Microwaves Antennas and Propagation vol 7no 6 pp 452ndash457 2013

[115] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[116] U Singh and T S Kamal ldquoSynthesis of thinned planarconcentric circular antenna arrays using biogeography-basedoptimisationrdquo IET Microwaves Antennas and Propagation vol6 no 7 pp 822ndash829 2012

[117] S K Goudos K B Baltzis K Siakavara T Samaras E Vafiadisand J N Sahalos ldquoReducing the number of elements in lineararrays using biogeography-based optimizationrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 1615ndash1618 Prague Czech Republic March2012

[118] X Li and M Yin ldquoHybrid differential evolution with bioge-ography-based optimization for design of a reconfigurableantenna arraywith discrete phase shiftersrdquo International Journalof Antennas and Propagation vol 2011 Article ID 685629 12pages 2011

[119] A Hoorfar S Lakhani and V Jamnejad ldquoApplication of levymutation operator in evolutionary programming optimizationof antennasrdquo in Proceedings of the International Conference onElectromagnetics in Advanced Applications (ICEAA rsquo07) pp 191ndash194 IEEE Torino Italy September 2007


[120] A Hoorfar and Y Liu ldquoAntenna optimization using an evolu-tionary programming algorithm with a hybrid mutation oper-atorrdquo in Proceedings of the in IEEE Antennas and PropagationSociety AP-S International Symposium (Digest) pp 1026ndash1029Salt Lake City UT USA July 2000

[121] R Sanchez-Montero S Salcedo-Sanz J A Portilla-Figuerasand R J Langley ldquoHybrid PIFA-patch antenna optimizedby evolutionary programmingrdquo Progress in ElectromagneticsResearch vol 108 pp 221ndash234 2010

[122] GACasulaGMazzarella andN Sirena ldquoEvolutionary designof wide-band parasitic dipole arraysrdquo IEEE Transactions onAntennas and Propagation vol 59 no 11 pp 4094ndash4102 2011

[123] N Hansen and A Ostermeier ldquoCompletely derandomized self-adaptation in evolution strategiesrdquo Evolutionary Computationvol 9 no 2 pp 159ndash195 2001

[124] N Hansen ldquoThe CMA evolution strategy a comparing reviewrdquoStudies in Fuzziness and Soft Computing vol 192 pp 75ndash1022006

[125] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII X Yao E K Burke J A Lozano et alEds vol 3242 of Lecture Notes in Computer Science pp 282ndash291 Springer 2004

[126] M D Gregory Z Bayraktar and D H Werner ldquoFast optimiza-tion of electromagnetic design problems using the covariancematrix adaptation evolutionary strategyrdquo IEEE Transactions onAntennas and Propagation vol 59 no 4 pp 1275ndash1285 2011

[127] M D Gregory S V Martin and D H Werner ldquoA comparisonof CMA-ES with other real-coded strategies used in electro-magnetics designrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium (APSURSI rsquo14) pp1962ndash1963 IEEE Memphis Tenn USA July 2014

[128] A A Al-Azza F J Harackiewicz and H R Gorla ldquoCovariancematrix adaptation evolutionary strategy optimization of patchantenna for wireless communicationrdquo Progress in Electromag-netics Research Letters vol 54 pp 85ndash91 2015

[129] X S Fang C K Chow K W Leung and E H LimldquoNew single-dual-mode design formulas of the rectangulardielectric resonator antenna using covariancematrix adaptationevolutionary strategyrdquo IEEE Antennas andWireless PropagationLetters vol 10 pp 734ndash737 2011

[130] S Haykin Neural Networks A Comprehensive FoundationMacmillan New York NY USA 1994

[131] C Christodoulou andMGeorgiopoulosApplications of NeuralNetworks in Electromagnetics Artech House Norwood MassUSA 2001

[132] Y Lecun Y Bengio and G Hinton ldquoDeep learningrdquo Naturevol 521 no 7553 pp 436ndash444 2015

[133] L Deng and D Yu ldquoDeep learning methods and applicationsrdquoFoundations and Trends in Signal Processing vol 7 no 3-4 pp197ndash391 2014

[134] I Arel D Rose and T Karnowski ldquoDeep machine learningmdashanew frontier in artificial intelligence researchrdquo IEEE Computa-tional Intelligence Magazine vol 5 no 4 pp 13ndash18 2010

[135] J Schmidhuber ldquoDeep learning in neural networks anoverviewrdquo Neural Networks vol 61 pp 85ndash117 2015

[136] S KGoudos ldquoDesign ofMicrowave broadband absorbers usinga self-adaptive differential evolution algorithmrdquo InternationalJournal of RF and Microwave Computer-Aided Engineering vol19 no 3 pp 364ndash372 2009

[137] H Ma D Simon M Fei and Z Chen ldquoOn the equivalencesand differences of evolutionary algorithmsrdquo Engineering Appli-cations of Artificial Intelligence vol 26 no 10 pp 2397ndash24072013

[138] L Cen Z L Yu W Ser and W Cen ldquoLinear aperiodicarray synthesis using an improved genetic algorithmrdquo IEEETransactions on Antennas and Propagation vol 60 no 2 pp895ndash902 2012

[139] ZD Zaharis andTV Yioultsis ldquoAnovel adaptive beamformingtechnique applied on linear antenna arrays using adaptivemutated Boolean PSOrdquo Progress in Electromagnetics Researchvol 117 pp 165ndash179 2011

[140] F Afshinmanesh A Marandi and M Shahabadi ldquoDesignof a single-feed dual-band dual-polarized printed microstripantenna using a Boolean particle swarm optimizationrdquo IEEETransactions on Antennas and Propagation vol 56 no 7 pp1845ndash1852 2008

[141] W-B Wang Q-Y Feng and D Liu ldquoApplication of chaoticparticle swarm optimization algorithm to pattern synthesis ofantenna arraysrdquo Progress in Electromagnetics Research vol 115pp 173ndash189 2011

[142] M Donelli A Martini and A Massa ldquoA hybrid approachbased on PSO and hadamard difference sets for the synthesisof square thinned arraysrdquo IEEE Transactions on Antennas andPropagation vol 57 no 8 pp 2491ndash2495 2009

[143] R Li L Xu X-W Shi N Zhang and Z-Q Lv ldquoImproved dif-ferential evolution strategy for antenna array pattern synthesisproblemsrdquo Progress in Electromagnetics Research vol 113 pp429ndash441 2011

[144] A Massa M Pastorino and A Randazzo ldquoOptimization of thedirectivity of a monopulse antenna with a subarray weightingby a hybrid differential evolution methodrdquo IEEE Antennas andWireless Propagation Letters vol 5 no 1 pp 155ndash158 2006

International Journal of

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RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Submit your manuscripts athttpwwwhindawicom

VLSI Design



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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Evolutionary algorithms (EAs)

GeneticAlgorithms

EvolutionaryProgramming

Evolutionstrategies

DifferentialEvolution

SwarmIntelligence

CMA-ES PSO ABC ACO

Figure 1 A diagram depicting main families of evolutionary algorithms

Free Lunchrdquo (NFL) theorem which pertains to the averagebehavior of optimization algorithms over given spaces ofoptimization problems It has been shown in [25] that whenaveraged over all possible optimization problems definedover some search space 119883 no algorithm has a performanceadvantage over any other Additionally in [26] the authorsshow that it is theoretically impossible to have a best general-purpose universal optimization strategy and the only way forone strategy to be superior to the others is when it focuseson a particular class of problems It should be noted that awide variety of optimization problems arise in the antennadomain and that it is not always an easy task to find the bestoptimization algorithm for solving each of them Thereforeit is worthwhile to explore new optimization algorithms ifwe find that they can work well for the problem at handOptimization problems arising in the design and synthesis ofantennas can benefit considerably from an application of theEAs which can lead to unconventional solutions regardingposition and excitation of the antenna elements in an arrayFurthermore the elements themselves can be geometricallydesigned by using the EAs

The purpose of this paper is to briefly describe thealgorithms and present their application to antenna designproblems found in the recent literature

Section 2 reviews the Genetic Algorithms (GAs) Sec-tion 3 focuses on PSO and Section 4 is devoted to DE A briefoverview of other EA algorithms is provided in Section 5We describe the algorithms briefly and then present somestatistics regarding the use of the algorithms in the literatureAdditionally for each algorithm some representative papersare referenced in the tables Some open issues are discussedin Section 6 and conclusions are drawn in Section 7

2 Genetic Algorithms

GAs the most popular EAs are inspired by Darwinrsquos naturalselection GAs can be real or binary-coded In a binary-coded

GA each chromosome encodes a binary string [27 28]The most commonly used operators are crossover mutationand selection The selection operator chooses two parentchromosomes from the current population according to aselection strategy Most popular selection strategies includeroulette wheel and tournament selection The crossoveroperator combines the two parent chromosomes in order toproduce one new child chromosome The mutation operatoris applied with a predefined mutation probability to a newchild chromosome

A search in the Scopus database shows that there are65762 conference papers and 94510 journal papers relatedto GAs from 1977 to 2016 Figure 2 shows the numberof papers related to GAs and antennas over the last 15years Additionally the search in the same database usingthe keywords GAs and Antennas reveals a total number of2807 papers (both journal and conference) Tutorials andapplications of GAs to electromagnetics can be found in[29 30]

Table 1 lists selected papers that use GAs (mostly binary-coded) for antenna design Several papers exist in the lit-erature that apply GAs for array synthesis A binary-codedGA has been applied for linear and planar array synthesis in[31 32] among others while a GA that uses decimal operatorhas been used in [33] The problem of array thinning hasbeen addressed in the literature using binary-coded GAs in[34] while the array-failure correction problem has beenaddressed by using the real-coded GAs in [35] A simple GAhas been used for the synthesis of time-modulated arrays in[36] Different antenna types have been designed using theGAs for example wire antennas in [37ndash40] patch antennasin [41 42] and RFID tags in [43] Other GA-type algorithmssuch as a mixed integer GA have also been applied in [44]to several different antenna design problems An improvedGA for the design of linear aperiodic arrays and a binary-coded GA which uses two-point crossover (instead of theusual one point) have been used in [45] for the design of








Pape

r num

ber

140

120

100

80

60

40

20

0

Year1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016







119906119866+1119899119894

= 119908119906119866119899119894

+ 1198881rand1(01)

(119901119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)

+ 1198882rand2(01) (119892119887119890119904119905119866+1119899119894 minus 119909119866119899119894)

(1)

where 119906119866+1119899119894



rand1(01)

rand2(01)



2








119906119866+1119899119894

= 119870 [119906119866119899119894

+ 1198881rand1(01)

(119901119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)

+ 1198882rand2(01)

(119892119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)]

119870 =2


1003816100381610038161003816100381610038161003816

(2)






Pape

r num

ber

Year2002 2004 2006 2008 2010 2012 2014 2016

0

20

40

60

80








119909119866119894

is computed by

DErand1bin

V119866+1119894 = 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) 1199031 = 1199032 = 1199033

DErand-to-best2bin

V119866+1119894

= 119909119866119894+ 119865 (119909

119866best minus 119909119866119894) + 119865 (1199091198661199031

minus 1199091198661199032

)

+ 119865 (1199091198661199033

minus 1199091198661199034

) 1199031

= 1199032

= 1199033

= 1199034

DErand2bin

V119866+1119894

= 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) + 119865 (1199091198661199034

minus 1199091198661199035

)

1199031

= 1199032

= 1199033

= 1199034

= 1199035

(3)



= (119906119866+11119894

119906119866+12119894

119906119866+1119895119894

119906119866+1119863119894


119906119866+1119895119894

=

V119866+1119895119894

if rand119895[01)

le CR or 119895 = rn (119894)

119909119866+1119895119894

if rand119895[01)

gt CR and 119895 = rn (119894) (4)

where 119895 = 1 2 119863 rand119895[01)



Year2002 2004 2006 2008 2010 2012 2014 2016

Pape

r num

ber

0

5

10

15

20

25

30

35



119909119866+1119894

=

119906119866+1119894

if 119891 (119906119866+1119894

) lt 119891 (119909119866119894)

119909119866119894 otherwise

(5)


119866+1119894




























7 Conclusion





Competing Interests


Acknowledgments


References
























































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Pape

r num

ber

140

120

100

80

60

40

20

0

Year1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016







119906119866+1119899119894

= 119908119906119866119899119894

+ 1198881rand1(01)

(119901119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)

+ 1198882rand2(01) (119892119887119890119904119905119866+1119899119894 minus 119909119866119899119894)

(1)

where 119906119866+1119899119894



rand1(01)

rand2(01)



2








119906119866+1119899119894

= 119870 [119906119866119899119894

+ 1198881rand1(01)

(119901119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)

+ 1198882rand2(01)

(119892119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)]

119870 =2


1003816100381610038161003816100381610038161003816

(2)






Pape

r num

ber

Year2002 2004 2006 2008 2010 2012 2014 2016

0

20

40

60

80








119909119866119894

is computed by

DErand1bin

V119866+1119894 = 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) 1199031 = 1199032 = 1199033

DErand-to-best2bin

V119866+1119894

= 119909119866119894+ 119865 (119909

119866best minus 119909119866119894) + 119865 (1199091198661199031

minus 1199091198661199032

)

+ 119865 (1199091198661199033

minus 1199091198661199034

) 1199031

= 1199032

= 1199033

= 1199034

DErand2bin

V119866+1119894

= 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) + 119865 (1199091198661199034

minus 1199091198661199035

)

1199031

= 1199032

= 1199033

= 1199034

= 1199035

(3)



= (119906119866+11119894

119906119866+12119894

119906119866+1119895119894

119906119866+1119863119894


119906119866+1119895119894

=

V119866+1119895119894

if rand119895[01)

le CR or 119895 = rn (119894)

119909119866+1119895119894

if rand119895[01)

gt CR and 119895 = rn (119894) (4)

where 119895 = 1 2 119863 rand119895[01)



Year2002 2004 2006 2008 2010 2012 2014 2016

Pape

r num

ber

0

5

10

15

20

25

30

35



119909119866+1119894

=

119906119866+1119894

if 119891 (119906119866+1119894

) lt 119891 (119909119866119894)

119909119866119894 otherwise

(5)


119866+1119894




























7 Conclusion





Competing Interests


Acknowledgments


References
























































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119906119866+1119899119894

= 119870 [119906119866119899119894

+ 1198881rand1(01)

(119901119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)

+ 1198882rand2(01)

(119892119887119890119904119905119866+1119899119894

minus 119909119866119899119894

)]

119870 =2


1003816100381610038161003816100381610038161003816

(2)






Pape

r num

ber

Year2002 2004 2006 2008 2010 2012 2014 2016

0

20

40

60

80








119909119866119894

is computed by

DErand1bin

V119866+1119894 = 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) 1199031 = 1199032 = 1199033

DErand-to-best2bin

V119866+1119894

= 119909119866119894+ 119865 (119909

119866best minus 119909119866119894) + 119865 (1199091198661199031

minus 1199091198661199032

)

+ 119865 (1199091198661199033

minus 1199091198661199034

) 1199031

= 1199032

= 1199033

= 1199034

DErand2bin

V119866+1119894

= 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) + 119865 (1199091198661199034

minus 1199091198661199035

)

1199031

= 1199032

= 1199033

= 1199034

= 1199035

(3)



= (119906119866+11119894

119906119866+12119894

119906119866+1119895119894

119906119866+1119863119894


119906119866+1119895119894

=

V119866+1119895119894

if rand119895[01)

le CR or 119895 = rn (119894)

119909119866+1119895119894

if rand119895[01)

gt CR and 119895 = rn (119894) (4)

where 119895 = 1 2 119863 rand119895[01)



Year2002 2004 2006 2008 2010 2012 2014 2016

Pape

r num

ber

0

5

10

15

20

25

30

35



119909119866+1119894

=

119906119866+1119894

if 119891 (119906119866+1119894

) lt 119891 (119909119866119894)

119909119866119894 otherwise

(5)


119866+1119894




























7 Conclusion





Competing Interests


Acknowledgments


References
























































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119909119866119894

is computed by

DErand1bin

V119866+1119894 = 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) 1199031 = 1199032 = 1199033

DErand-to-best2bin

V119866+1119894

= 119909119866119894+ 119865 (119909

119866best minus 119909119866119894) + 119865 (1199091198661199031

minus 1199091198661199032

)

+ 119865 (1199091198661199033

minus 1199091198661199034

) 1199031

= 1199032

= 1199033

= 1199034

DErand2bin

V119866+1119894

= 1199091198661199031

+ 119865 (1199091198661199032

minus 1199091198661199033

) + 119865 (1199091198661199034

minus 1199091198661199035

)

1199031

= 1199032

= 1199033

= 1199034

= 1199035

(3)



= (119906119866+11119894

119906119866+12119894

119906119866+1119895119894

119906119866+1119863119894


119906119866+1119895119894

=

V119866+1119895119894

if rand119895[01)

le CR or 119895 = rn (119894)

119909119866+1119895119894

if rand119895[01)

gt CR and 119895 = rn (119894) (4)

where 119895 = 1 2 119863 rand119895[01)



Year2002 2004 2006 2008 2010 2012 2014 2016

Pape

r num

ber

0

5

10

15

20

25

30

35



119909119866+1119894

=

119906119866+1119894

if 119891 (119906119866+1119894

) lt 119891 (119909119866119894)

119909119866119894 otherwise

(5)


119866+1119894




























7 Conclusion





Competing Interests


Acknowledgments


References
























































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
































7 Conclusion





Competing Interests


Acknowledgments


References
























































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















7 Conclusion





Competing Interests


Acknowledgments


References
























































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Competing Interests


Acknowledgments


References
























































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Review Article Evolutionary Algorithms Applied to Antennas ...Review Article Evolutionary Algorithms Applied to Antennas and Propagation: A Review of State of the Art SotiriosK.Goudos,