Self-AdaptiveDifferentialEvolutionAlgorithmWithZoningEvolutionofControlParameters

andAdaptiveMutationStrategies

QinqinFanandXuefengYan

Abstract—Theperformanceofthedifferentialevolution(DE)algorithmissigni?cantlyaffectedbythechoiceofmutationstrategiesandcontrolparameters.Maintainingthesearchcapa-bilityofvariouscontrolparametercombinationsthroughouttheentireevolutionprocessisalsoakeyissue.Aself-adaptiveDEalgorithmwithzoningevolutionofcontrolparametersandadap-tivemutationstrategiesisproposedinthispaper.Intheproposedalgorithm,themutationstrategiesareautomaticallyadjustedwithpopulationevolution,andthecontrolparametersevolveintheirownzoningtoself-adaptanddiscovernearoptimalval-uesautonomously.Theproposedalgorithmiscomparedwith?vestate-of-the-artDEalgorithmvariantsaccordingtoasetofbenchmarktestfunctions.Furthermore,sevennonparamet-ricstatisticaltestsareimplementedtoanalyzetheexperimentalresults.Theresultsindicatethattheoverallperformanceoftheproposedalgorithmisbetterthanthoseofthe?veexistingimprovedalgorithms.

IndexTerms—Controlparameteradaptation,differentialevolution(DE)algorithm,mutationstrategyadaptation,zoningevolution.

I.INTRODUCTION

IFFERENTIALevolution(DE)algorithmproposedbyStornandPrice[1]isacompetitiveandreliableevo-lutionarycomputingtechniqueforsolvingawidevari-etyofcomplexoptimizationproblems.Theoptimizationperformance[2]–[5]oftheDEalgorithmnotonlydependsonthechoiceofthreecontrolparameters(i.e.,muta-tioncontrolparameterF,crossovercontrolparameterCR,andpopulationsizeNP),butalsoonthechoiceoftrialvectorgenerationstrategies(i.e.,mutationandcrossoverstrategies).Toimprovethealgorithm’sperformance,severalusefulempiricalguidelinesforselectingcontrolparametersettingsandmutationstrategieshavebeenintroducedbymanyresearchersduringthepastdecade.Eibenetal.[6]and

ManuscriptreceivedDecember10,2013;revisedNovember16,2014andJanuary21,2015;acceptedJanuary27,2015.DateofpublicationMarch10,2015;dateofcurrentversionDecember14,2015.Thisworkwassupportedinpartbythe973projectofChinaunderGrant2013CB733600,inpartbytheNationalNaturalScienceFoundationofChinaunderGrant21176073,inpartbytheProgramforNewCenturyExcellentTalentsinUniversityunderGrantNCET-09-0346,andinpartbytheFundamentalResearchFundsfortheCentralUniversities.ThispaperwasrecommendedbyY.Jin.

TheauthorsarewiththeKeyLaboratoryofAdvancedControlandOptimizationforChemicalProcessesofMinistryofEducation,EastChinaUniversityofScienceandTechnology,Shanghai200237,China(e-mail:xfyan@ecust.edu.cn).

Colorversionsofoneormoreofthe?guresinthispaperareavailableonlineathttp://ieeexplore.ieee.org.

DigitalObjectIdenti?er10.1109/TCYB.2015.2399478

D

Brestetal.[7]proposedthreeparametercontroltechniques(i.e.,deterministic,adaptive,andself-adaptive)andanencod-ingtechnique(i.e.,encodingcontrolparametersFandCRintoanindividual),respectively.However,differentoptimizationproblemsorparticularevolutionstagesoftenrequirechosenmutationstrategiesandsuitablecontrolparameters[8]dur-ingtheevolutionprocess;aninappropriatechoiceofmutationstrategiesandcontrolparametersmaydirectlyin?uencethealgorithm’sperformance[9]–[11].Differentcontrolparam-etercombinationsmayalsosigni?cantlyaffectalgorithm’ssearchcapability.Wangetal.[5]andMallipeddietal.[8]employedconstantcombinationsofcontrolparameters(i.e.,FandCRvaluesaredeterminedbeforeactualDEusage)tomaintainthesearchingcapabilityofthecontrolparame-tercombinations;however,theirproposedmethodsmaybeunabletoadaptquicklyincomplexoptimizationenvironments.Onthebasisoftheseconsiderations,obtainingthemostsuitablemutationstrategyandreasonablecombinationsofcontrolparametersatdifferentevolutionphasesisimportanttoimproveDEperformance.Inthispaper,aself-adaptiveDE(SDE)algorithmwithzoningevolutionofcontrolparame-tersandadaptivemutationstrategies(ZEPDE)isproposed.InZEPDE,thenumberofeachmutationstrategycanbegraduallyadjustedviaaroulettewheel,andreal-timeoptimalcontrolparametercombinationscanbeobtainedbyzoningevolution,whichcanmaintainthecontrolparameterdistribution.ZEPDEwascomparedwith?veimprovedDEvariantsaccordingto25CEC2005and28CEC2013testfunctions.

Theremainderofthispaperisorganizedasfollows.SectionIIintroducesthebasicDEalgorithm.SectionIIIreviewstherelatedresearchesonDEalgorithm.SectionIVpresentstheproposedZEPDEalgorithm.SectionVreportstheexperimentalresultsandsensitiveanalysisofZEPDEparameters.Finally,theconclusionissummarizedinSectionVI.Togainbetterunderstanding,interestedreaderscanrefertothesupplementary?le.

II.DEALGORITHM

IntheevolutionprocessofDE,mutation,crossover,andselectionoperatorsareperformed.Thevectorcon-tainingDoptimizedvariablesx1,x2,...,xDisdenoted

GGGbyx.xGi=[xi,1,xi,2,...,xi,D]denotestheithsolution

(orindividual)intheGthgeneration.Thepopulationofthe

GGGthgenerationisdenotedbyXG=[xG1,x2,...,xNP],which

c2015IEEE.Personaluseispermitted,butrepublication/redistributionrequiresIEEEpermission.2168-2267??

Seehttp://www.ieee.org/publications_standards/publications/rights/index.htmlformoreinformation.

220containsNPindividuals.NPgenerallyis?xedduringtheevolutionprocess.Theminimumproblemisdescribedasfollows:

minx

fx??

(x1,x2,...,xj∈xlowj,xhigh

??D)j

,j=1,2,...,D(1)

wherefdenotesthefunctionthatissubjecttooptimization.AD-dimensionalspaceP0isde?nedwithintheregion{(xlowj,xhighj

)j=1,2,...,D}.TheprocedureofexecutingDEisasfollows[9].1)InitializationOperation:Theinitialindividualsx0NParerandomlygeneratedinPi,i=1,2,...,0.MutationF,crossoverCR,andmaximumnumberofgenerationsGmaxaredetermined.ThecurrentgenerationissetasG=0.

2)ForeachindividualxGi,i=1,2,...,NP,performsteps3)–5)toderivethepopulationforthenextgeneration.3)MutationOperation[12]:ForeachxGpopulation,themutantindividualx?Giinthei+1

parent

isgeneratedasfollows:

x?Gi+1=xG??

??rGrG

1+F·x2?xr3(2)wherer1,r2,r3∈{1,2,...,NP}arerandomlychosen

andaredifferentfromtherunningindexi.Fisarealconstantscalingfactorwithin[0,1],whichcontrolsthe

ampli?cationofthedifferentialvariation(xGrGOperation,[9]:ForeachxG2?xr3).

4)CrossoverxˉGi,ai+1trialindividual

isgenerated??asfollows:

xˉGx?Gij+1,Rij

+1

=j≤CRxGj=1,2,...,D(3)ij,otherwiseofwhere[0,1].

Rjisauniformrandomnumberintherange

5)EvaluationOperation,[9]:OffspringxˉG+1

competesone-to-onewithitsparentxGevaluationi

i.Theoperation

isexpressedas

??

xGi+1=xˉGi+1,f??xˉGi+1??≤f(xGi)xGi,otherwise.(4)6)G=G+1.

7)Steps2)–6)arerepeatedaslongasthenumberofgenerationsissmallerthantheallowablemaximumnumberGmax.

Otherusefulstrategies“DE/rand/2”[13]:x?Gi+1areasfollows:

=xG“DE/best/2”[12]:x?Gr1+F·(xGr2?xGGGi+1=xGbest+F·(xGxrr1?G3)+F·(xrr2)+F·(xG4?xrr3?xG5)r4)

“DE/current-to-best/1”[12]:

x?Gi+1=xG??

??i+F·xGbest?xG??i+F·xGr1?xG

??r2“DE/current-to-best/2”[13]:

x?Gi

+1=xG??

??????best+F·xGbest?xGi+F·xGr1?xGr2+xGr3?xG

r4wherexGbestistheindividualvectorwiththebest?tnessvalueinthepopulationatgenerationG.

IEEETRANSACTIONSONCYBERNETICS,VOL.46,NO.1,JANUARY2016

III.LITERATUREREVIEW

AlthoughtheDEalgorithmhasattractedtheattentionof

manyresearchersbecauseofitshighsearchaccuracy,robust-ness,andgoodconvergencespeed,selectingsuitablemutationstrategiesandcontrolparametersatdifferentevolutionstagesisdif?cult,particularlywhentheoptimizationproblemsarecomplexandrequiredifferentsearchcapabilities.

ToimproveDEperformance,DEresearchershavepro-posedseveralempiricalguidelinesforchoosingcontrolparam-etersandmutationstrategiesoverthepastdecade.Forinstance,StornandPrice[14]suggestedthatthesuitablepop-ulationsizeshouldbebetween5Dand10D;Fcanbewithintherangeof[0.4–1],andCRcanbe0.1or0.9.Basedontheexperimentalresultsforthreetestfunctions,Gamperleetal.[15]indicatedthattheappropriatepopula-tionsizeshouldbebetween3Dand8D;F=0.6canbeselected,andCRintherangeof[0.3–0.9]isagoodchoice.Ronkkonenetal.[16]recommendedthattherangeofpopu-lationsizebebetween2Dand4D.F=0.9wouldbeagoodinitialchoice,butasuitableFshouldbeselectedwithintherangeof[0.4–0.95].CRshouldbewithin[0–0.2]forsepa-rabletestfunctionsandwithin[0.9–1]formultimodalandparameterdependenttestfunctions.Zielinskietal.[17]sug-gestedthatthesettingsF≥0.6andCR≥0.6arebene?cialtoalgorithmconvergenceinmostcases.Moreover,severalresearchers[14],[15],[18]–[21]haveproposedandinvesti-gatedvariousmutationstrategiesinconsiderationofthefactthatdifferentoptimizationproblemsrequiredifferentsearchcapabilities.AlthoughtheseguidelinesareusefultoimproveDEperformance,theymaycauseconfusiontoDEuserswhenthepropertiesofoptimizationproblemsarenotsuf?cientlyclearandlacksuf?cientjusti?cationsbecausetheseguidelinesweredesignedbasedonspeci?cexperiments(i.e.,lackofuni-versality)[13].ConstantcontrolparametersettingsandsinglemutationstrategyalsoviolatethenatureofDarwinianevolu-tionandcannotadapttodifferentoptimizationproblemsandevolutionphases.

Manyadaptiveorself-adaptiveDEvariantshavebeenproposedtoavoidmanualtuningofDEcontrolparame-tersandtheuseofaconstantmutationstrategyduringtheevolutionprocess.Forexample,Abbass[22]introducedaself-adaptiveparetodifferentialevolutionalgorithm(SPDE)whereinthecontrolparametersFandCRaregeneratedbyusingaGaussiandistribution.OnthebasisoftheDEalgorithm,Vuetal.[23]recentlyintroducedadirection-guidedevolutionalgorithm(DEAL)whereintwotypesofdirections(i.e.,convergenceandspreadingdirections)areuti-lizedtoguidethepopulationevolution.TheirexperimentalresultsshowthatDEALcanperformbetterthanotherwell-knownalgorithms.LiuandLampinen[11]proposedafuzzyadaptiveDE(FADE)algorithmwhereinappropriatecontrolparametersFandCRcanbegeneratedbyusingfuzzylogiccontrollers.TheirresultsindicatethattheperformanceofFADEinasetofbenchmarksetfunctionsisbetterthanthatofstandardDEalgorithms.ZhangandSanderson[21]pro-posedanewDEalgorithm(JADE),inwhichamutationstrategywithanoptionalexternalarchiveisusedandcon-trolparametersFandCRcanbeautomaticallyadjustedbased

FANANDYAN:SDEALGORITHMWITHZEPDETABLEI

INDIVIDUALWITHCONTROLPARAMETERSANDMUTATIONSTRATEGY

ontheirpreviousrecordofsuccess.Zaharie[24]introducedaparameteradaptationforDE(ADE)inwhichtheadjust-mentofFandCRisbasedonpopulationdiversityandamultipopulationapproachisused.Brestetal.[7]proposedaself-adaptivejDE,inwhicheachindividualhasitsowncontrolparameters(i.e.,FandCR).Thecontrolparameterswereadjustedbytwonewparameters.Omranetal.[25]introducedaSDEwhereinFisadaptiveandCRisgener-atedfromanormaldistributionN(0.5,0.15).Qinetal.[13]alsointroducedaSDEwhereinbothmutationstrategiesandtheirassociatedcontrolparameterscangraduallyself-adaptbylearningfromtheirprevioussuccessfulexperiences.ResultsindicatethatSaDEoutperformsseveralDEvariantsonasetof26testfunctions.Panetal.[26]andMallipeddietal.[8]proposedaDEalgorithmwithaself-adaptivetrialvectorgenerationstrategyandcontrolparameters(SspDE)andanensembleofmutationstrategiesandcontrolparameterswithDE(EPSDE),respectively.InbothDEvariants,mutationstrategiesandcontrolparameterscanautomaticallyadapttopopulationevolution.Ghoshetal.[27]introducedanimprovedDE(FiADE)whereintheadaptationofFandCRisbasedonthe?tnessfunctionvaluesofindividuals.TheirexperimentalresultsshowthatFiADEisacompetitiveoptimizationtoolforsolvingvariousoptimizationproblems.Wangetal.[5]introducedacompositeDE(CoDE),inwhicheachmutationstrategyisrandomlycombinedwitha?xedcontrolparame-tersetting.CoDEemploysthreemutationstrategiesandthreecontrolparametersettings.Gongetal.[28]introducedanenhancedDEinwhichanewstrategyadaptationmechanismisused.Meanwhile,severalresearchershaveappliedanadaptiveorself-adaptivepopulationsizeinDE.Teo[29],[30]proposedthe?rstaself-adaptivepopulationsizeapproach;absoluteandrelativeencodingmethodologiesareemployedinthepopu-lationsizeoftheproposedalgorithm.Tengetal.[31]alsoutilizedtwodifferentencodingmethodologiestoimplementaself-adaptivepopulationsize.TirronenandNeri[32]pro-posedanadaptivepopulationsizewhereinameasurementofthe?tnessdiversitywasused.Zhuetal.[33]introducedanadaptivepopulationtuningschemetoadjustpopulationsize.Variousotherpopulationsizereductionschemeshavebeenproposedby[34],[35].

TheDEalgorithmhasbeensuccessfullyappliedtosolvevariousreal-worldengineeringoptimizationproblemsduetoitseffectiveness,robustness,andsimplicity.Forexample,ZhongandZhang[36]proposedanadaptiveDEalgorithmtosolvesubpixelmapping.Chenetal.[37]introducedamodi?eddifferentialevolutionwhichisappliedtoobtaintheoptimalcontrollerparametersofanadaptiveneuralfuzzynetwork.

221

Fig.1.ZoningtypeofZEPDE.

NeriandMininno[38]proposedamemeticcompactdif-ferentialevolutionforthecontrolofcommercialrobots.Zamudaetal.[39]proposedanapproachbasedontheDEalgorithmforprocedural3-Dmodelsreconstructionofwoodyplants.Salvatoreetal.[40]employedtheDEalgorithmtoopti-mizeanalgorithmofsensorlesscontrolofinductionmotors.Aminetal.[41]employedtheDEalgorithmtoadjusttheairtraf?ccontrollerstaskloadinrealtime.

IV.SELF-ADAPTIVEDIFFERENETIALEVOLUTION

ALGORITHMWITHZEPDEAlthoughmanyself-adaptiveDEalgorithmshavebeenintroduced,howtomaintainthesearchcapabilitiesofcon-trolparametercombinationsofdifferentzoning(i.e.,zoningevolutionofcontrolparameters)duringtheevolutionprocesshasnotbeenconsideredbyotherDEresearchers.Forexam-ple,EPSDEandCoDEemploy?xedcombinationsofcontrolparametersthroughouttheentiresearchprocess.Self-adaptivecontrolparametersaregeneratedinSaDEandJADE,buttheperformanceofcontrolparametercombinationsindifferentzoningisnotconsidered.Basedontheseobservations,wepro-posetheZEPDEalgorithm,inwhichtheappropriatemutationstrategycanbegraduallyadjustedbyaroulettewheel,andsuitablecombinationsofFandCRcanbegeneratedbyzon-ingevolution.Eachindividualhasitsowncontrolparametercombinationandmutationstrategy(TableI).

A.Self-AdaptiveControlParametersBasedonZoningEvolution

ToachievereasonablecombinationsofcontrolparametersduringdifferentevolutionstagesinZEPDE,thetotalregionofcontrolparameters(i.e.,FandCR)isdividedintofoursame-sizeareas(Fig.1).Thezoningevolutionoperationsofcontrolparametersoftheproposedalgorithmcanbedescribedbythefollowingsteps.

1)Countthenumberofcontrolparameterscombina-tionsineachregion.Thenumberofcontrolparam-etercombinationsbeNGinthehthregionisassumedto

,h=1,2,...,H,whereHisthetotalnumberofzoningh

(NZ).

2)Ineachzoning,ifoffspringindividual?tnessisbet-terthanthatofitsparent,thenitscontrolparam-etercombinationisregardedastheelitecontrol

222Fig.2.

Procedureofzoningevolution.

parametercombination(EPC).IfthehthregionhasEPC,assumingthatthenumberofEPCinisNGh,elite,0≤NGh,elite≤NGthehthregion

h,thentheweightedvalueofeachEPCinthehthregioniscomputedasfollows:

??

??????fxˉG+1wGNGh,e?fxGNG

hNGh,e=??NG??????,e??(5)h,elitefe=1

xˉG+1NGh,NG

h,e

?fxGNGh,e

whereNGhthh,e=1,2,...,NGregiondoesnoth,elite.

IfthehaveEPC,theneachcontrolparametercombinationisregardedasEPC.Welet

wGN

Gh,e

=1/NG

h.(6)

3)TheweightedaveragevaluesofFandCRinthehthregionarecomputedasfollows:

NGh,elite

FG

W,h

=

??

wGNG

×G

(7)

NG=1h,e

FNGh,e

h,eNGh,elite

CRGw,h=

??

wGNGNGh,e

×CRGNG

h,e

.(8)

h,e

=14)Thecontrolparametersaregeneratedinthehthregion

asthefollowingway:

FGh,+????

l1=CauchyFGW,h,l=1,2,...,NG

h(9)CRGh,+l1=N??CRG??

σ

w,h,σl=1,2,...,NGh(10)whereCauchyandNaretheCauchyandnormaldis-tributionfunctions,respectively.σ=0.55?0.3×(1?G/Gmax).

ThezoningevolutionofcontrolparametersisshowninFig.2.

IEEETRANSACTIONSONCYBERNETICS,VOL.46,NO.1,JANUARY2016

B.Self-AdaptiveMutationStrategies

TheDE/rand/1mutationstrategyhasgoodexplorationcapa-bilityandiswidelyutilizedinDE.Meanwhile,self-adaptivemutationstrategies(whichcontaingreedymutationstrategies)mayleadtodecreasedpopulationdiversityduringtheearlyevolutionprocess.Toobtaingoodbalancebetweenglobalandlocalsearchcapabilitiesintheproposedalgorithm,theDE/rand/1strategyisusedtooptimizethepopulationduringtheearlystagesoftheevolutionprocess.Subsequently,self-adaptivemutationstrategiesareutilizedtosearchfortheopti-malsolutioninconsiderationofthefactthatdifferentevolutionstagesrequiredifferentsearchcapabilities.OnthebasisofthemutationstrategiesselectedinSaDE[13]andCoDE[5],?vemutationstrategiesareselectedfortheproposedalgorithm.These?vestrategiesareDE/rand/1,DE/rand/2,DE/best/2,DE/current-to-best/1,andDE/current-to-best/2.

C.BoundaryOperationofControlParameters

AnextremelylargevalueofFcanincreasepopulation

diversityandleadtoslowconvergencespeed,whereasanextremelysmallvalueofFmayleadtostagnationorpre-matureconvergence[4].Therefore,therangeofFinthispaperiswithin[0.1–1](similartojDEin[7])duringtheearlyevolutionprocessandwithin[0–1]duringthelaterstagesofevolution.Meanwhile,therangeofCRisfrom[0–1]through-outtheentiresearchprocess.InfeasiblecontrolparametersareresetviatheweightedaveragevaluesofFandCRintheearlystagesofevolutionandregeneratedontheboundaryinthemiddleandlatestagesbecauseCauchyandnormaldistribu-tionfunctionsmayresultinthelossofboundarydistributionofthecontrolparameters.

D.OverallImplementationofZEPDE

Theproposedalgorithmisdescribedasfollows.

1)InitializationOperation:ThevaluesofparameterssuchasthenumberofindividualsinthepopulationNP,max-imumvariationofselectiveprobability(Msp)ineachgeneration,andmaximumgenerationsGmaxaredeter-mined.SetSetp=0.175(i.e.,onlyuseDE/rand/1muta-tionstrategybeforeGs=Setp×Gmaxgenerations),currentgenerationG=0,andBset=0.35(i.e.,abound-aryoperationofcontrolparametersisusedbeforeGb=Bset×Gmaxgenerations,whereastheotherboundaryoperationisemployedafterGb=Bset×Gmaxgener-ations).TheinitialpopulationP01isgenerated,andthenumberofindividualsegyisset(i.e.,NGbelongingtoeachmutationGstrat-str_randsG/1=Nstr_rands/2=Ns

NGstr_best/2=str_sGcurrent-to-best/1=Nstr_scurrentto-best/2=NP/5).

2)PopulationEvolution(MutationOperation):ForeachindividualxGi,i=1,2,...,NP,ifGtrialindividualx?Gi+1

isgeneratedbyusingthecorre-spondingmutationstrategy,whichisrandomlyselectedfromthemutationstrategypool.

FANANDYAN:SDEALGORITHMWITHZEPDEFig.3.FrameworkofZEPDE.

Boundaryoperation:Ifx?Ghigh

ij+1x,

thenx?Gj

ij+1=xG.Crossoveroperation

r

1j

??G+1xˉGx?ij,Rj≤CRij+1

=xGij

,otherwisej=1,2,...,D.Selectionoperation

??

????

xGi+1=

xˉGi+1,fxˉG+1

≤f??i

xG??

i

xGi,

otherwise.

3)Self-AdaptiveMutationStrategy:IfG=Gs,?vemuta-tionoperations,i.e.,DE/rand/1,DE/current-to-best/2,

DE/current-to-best/1,DE/best/2,andDE/rand/2,areran-domlyallocatedforallindividuals.EachindividualhasitsownmutationationhasNGoperation,andeachmutationoper-str_randsG/1=Nstr_randsG/2=Nstr_bests

/2=NGstr_current-to-bestsGG≥G/1=Nstr_current-to-bests/2=NP/5individ-uals.Ifs,themaximumobjectivefunctionvalueisobtainedasfollows:

f????

max=maxfxˉG????i+1

,i=1,2,...,NP.(11)Thedifferencebetweentheobjectivefunctionvalueof

eachindividualandfmaxiscalculatedinthefollowingway:

??fG+1

??????i

=??fxˉG??

??i+1?fmax????,i=1,2,...,NP.(12)223

Foreachmutationstrategy,thesumofdifferenceiscomputedas

NGstr_name

SG+1

=??

str_name

??fG+1

str_name,k,k=1,2,...,NG

str_name

k=1

(13)

wherestr_namegiesandNGdenotesoneofthe?vemutationstrate-denotesthenumberofindividualsforaspecialmutationstr_name

strategy.

Thesumofdifferenceofthe?vemutationstrategiesiscalculatedasfollows:

SG+1=SGrand+1/1+SGrand+1/2+SGcurrent-to-best/1

+1

+SGcurrent-to-best/2+1+SGbest+1

/2.

(14)

Theselectiveprobability(sp)ofeachmutationstrategy

iscomputedinthefollowing:

spGstr_name+1=SGstr_name+1/S

G+1

.(15)

Thespofeachmutationstrategycanbeformulatedas

follows:

???spG+Msp,ifspGstr_name+1

?spGstr_name>MspspGstr_name

+1=?str_name

??spG?str_name?Msp,ifspGstr_name+1?spGstr_name,

otherwise.(16)

Thecumulativeprobability(cp)ofeachmutationstrat-egyiscomputedasfollows:

cpGrand+1/1=spGrand+1

/1

(17)cpGrand+1/2=spGrand+1/1+spGrand+1/2

(18)cpGcurrent-to-best/1+1=spGrand+1/1+spGrand+1

/2

+spGcurrent-to-best/1

+1

(19)cpGcurrent-to-best/2+1=spGrand+1/1+spGrand+1/2+spGcurrent-to-best/1

+1

+spGcurrent-to-best/2

+1

(20)cpGbest+1/2=spGrand+1/1+spGrand+1/2+spGcurrent-to-best/1

+1

+spGcurrent-to-best/2+1+spGbest+1/2.

(21)

Basedonthecpofeachmutationberofeachmutationstrategy(i.e.,NGstrategy,+1theG+1

num-NGstr_+1current-to-best/1,NGstr_+1str_rand/1,Ncurrent-to-best/2,andNGwheel.

str_best+str_rand1/2,/2)isgeneratedbyaroulette4)Self-AdaptiveControlParameters:Thedetaileddescrip-tionsofthezoningevolutionoperationsofcontrolparametersarepresentedinSectionIV-A.

IfG<0.35×Gmax,FiG+1>1,orFasiG+1<0.1,themutationcontrolparameterisupdatedfollows:

FiG+1=N(μF,0.2)

whereμF=(FGW,A+FGW,BIfG<0.35×Gmax,CRG++1FG>W,G1C+ForCRW,DG)/+1

4.

<0,thecrossovercontrolparametericanbeupdatedi

asfollows:

CRGi

+1

=N(μCR,0.2)224IEEETRANSACTIONSONCYBERNETICS,VOL.46,NO.1,JANUARY2016

TABLEII

RESULTSOFWILCOXONTESTON30-DIMENSIONAL

CEC2005FUNCTIONSTABLEIII

RANKINGOBTAINEDBYFRIEDMAN’STESTON30-DIMENSIONAL

CEC2005FUNCTIONS

GGG

whereμCR=(CRGW,A+CRW,B+CRW,C+CRW,D)/4.IfG≥0.35×Gmax,themutationandcrossovercontrolparameterscanbeupdatedasfollows:

TABLEIV

p-VALUESOBTAINEDBYBONFERRONI–DUNN’S,HOLM’S,ANDHOCHBERG’SPROCEDURESFORTHECOMPAREDDEALGORITHMS

ON30-DIMENSIONALCEC2005FUNCTIONS

??

FiG+1

=??

+1CRGi

1,ifFiG+1>1,

????????????

??N0,0.15×1?(G/Gmax)2??,ifFG+1<0.

i

(22)

+1

1,ifCRG>1,i????????????

??N0,0.15×1?(G/Gmax)2??,ifCRG+1<0.

i

=

(23)

5)G=G+1.

6)Repeatsteps2)–5)aslongasthenumberofgenerationsisequaltotheallowablemaximumnumberGmax.TheframeworkofZEPDEisshowninFig.3.

V.EXPERIMENTALSTUDYONZEPDE

Todemonstratetheoverallperformanceoftheproposedalgorithm,25CEC2005[42]and28CEC2013[43]bench-markfunctionswereusedintheexperiments.TheoptimizationperformanceofZEPDEwasthencomparedwiththatof?vestate-of-the-artDEvariants,i.e.,jDE[7],SaDE[13],JADE[21],EPSDE[8],andCoDE[5].AlltheseDEvari-antswereprogrammedinMATLAB(MATLABR2012a)andrunonaWindows7operatingsystem(bit).Themaximumnumberoffunctionevaluations(FEs)issetto300000for30-dimensionaltestfunctionsand500000for50-dimensionaltestfunctionsinallalgorithms.Thenumberofindependentrunsissetto30andtheaccuracylevelofoptimizationresultsissettobe1E-8(i.e.,theoptimizationresultlessthan1E-8aszero).ThepopulationsizesettingsoftheseDEvariantsaredifferent,100forZEPDE,andthepopulationsizesettingsoftheotheralgorithmsaresimilartothoseintheirorigi-nalpapers,namely,100forjDEandJADE;50forEPSDEandSaDE;30forCoDE.Toeffectivelyanalyzetheexper-imentalresultsobtainedbythecomparedalgorithms,sevennonparametricstatisticaltestswithasigni?cancelevelof0.05areutilizedintheexperiments.ThesetestsareWilcoxon’sranksumtest[44],Kruskal–Wallistestwithmultiplecomparisons[45],Friedman’stest[46],Bonferroni–Dunn’stest[47],Holm’sprocedure,Iman–Davenporttest,andHochberg’sprocedure[48].FortheWilcoxon’sranksumtest,the“+,”“?,”and“≈”marksindicatethatZEPDEperformssigni?cantlybetterthan,worsethan,andalmostsimilartothecomparedalgorithms,respectively.FortheKruskal–Wallistestwithmultiplecomparisons,“K+”denotesthattheresultisthebestamongallcomparedresults,and“K?”or“K≈”indicate

thattheresultissigni?cantlyworsethanoralmostthesameasthebestresult,respectively.

A.ComparisonWithFiveImprovedDEon30-DimensionalProblems(CEC2005)

Inthisexperiment,ZEPDEiscomparedwith?vefamousDEvariants(i.e.,jDE,SaDE,JADE,EPSDE,andCoDE)accordingto2530-dimensionalCEC2005benchmarkfunc-tionstoevaluateitsoverallperformance.Theexperimen-talresultsaregiveninsupplementary(TableA1)duetopageslimit.

WilcoxontestisemployedtoanalyzetheoptimizationresultsshowninTableII.ItcanbeseenfromTableIIthatZEPDEperformsbetterthanjDE,SaDE,JADE,EPSDE,andCoDEon16,15,9,15,and7testfunctions,respectively.jDE,JADE,EPSDE,andCoDEperformsigni?cantlybetterthanZEPDEonone,?ve,?ve,andfourtestfunctions,respectively.TheperformanceofSaDEcannotbesigni?cantlybetterthanthatofZEPDEonanytestfunctions.

Friedman’stestisalsoemployedtoevaluatetheperfor-mancesofallthecomparedalgorithms.TherankingsobtainedbyFriedman’stestareshowninTableIII.TableIIIindi-catesthattheoverallperformanceofZEPDEperformsbetterthanthoseoftheothercomparedDEalgorithms.Todetectthedifferencesamongthecomparedalgorithms,p-valuesareobtainedbyBonferroni–Dunn’s,Holm’s,andHochberg’spro-ceduresshowninTableIV.TableIVindicatesthattheaverageperformanceofZEPDEissigni?cantlybetterthanthoseofjDE,EPSDE,andSaDEandissimilartothoseofJADEandCoDE.However,theresultsobtainedbyWilcoxontestandFriedman’stest,itisobservedthattheoverallperformanceofZEPDEisbetterthanthoseofJADEandCoDE.

B.ComparisonWithFiveImprovedDEon50-DimensionalProblems(CEC2005)

Inthisexperiment,2550-dimensionalCEC2005bench-markfunctionsareusedtotesttheperformancesofall

FANANDYAN:SDEALGORITHMWITHZEPDETABLEV

OPTIMIZATIONRESULTSFOR2550-DIMENSIONAL

CEC2005TESTFUNCTIONSTABLEVI

RANKINGOBTAINEDBYFRIEDMAN’STESTON50-DIMENSIONAL

CEC2005FUNCTIONS

TABLEVII

p-VALUESOBTAINEDBYBONFERRONI–DUNN’S,HOLM’S,ANDHOCHBERG’SPROCEDURESFORTHECOMPAREDDEALGORITHMS

ON50-DIMENSIONALCEC2005FUNCTIONS

comparedalgorithms.Theexperimentalresultsaregiveninsupplementary(TableA2)duetopageslimit.

TheWilcoxontestandFriedman’stestareemployedtodemonstratethedifferencesbetweenZEPDEandtheothercomparedalgorithms.TableVindicatesthatZEPDEsignif-icantlyoutperformsjDE,SaDE,JADE,EPSDE,andCoDEon16,20,11,18,and30testfunctions,respectively.TableVIshowsthattheoverallperformanceoftheproposedalgorithmisthebestamongthesecomparedalgorithms.

Bonferroni–Dunn’s,Holm’s,andHochberg’sproceduresareemployedtodetecttheglobaldifferencesamongthedifferentDEvariants.TableVIIshowsthattheaverageper-formanceofZEPDEissigni?cantlybetterthanthoseofjDE,EPSDE,andSaDEandissimilartothoseofJADEandCoDE.However,itcanbeobservedfromthestatisticalanalysisresultsthattheaverageperformanceoftheproposedalgorithmisthebestamongthecomparedalgorithmson2550-dimensionaltestfunctions.

C.ComparisonWithFiveImprovedDEon30-DimensionalProblems(CEC2013)

Inthisexperiment,2830-dimensionalCEC2013benchmarkfunctionsareusedtotesttheperformancesofallcomparedDEvariants.Theexperimentalresultsaregiveninsupplementary(TableA3)duetopageslimit.

225

TABLEVIII

RESULTSOFWILCOXONTESTON30-DIMENSIONAL

CEC2013FUNCTIONS

TABLEIX

RANKINGOBTAINEDBYFRIEDMAN’STESTON30-DIMENSIONAL

CEC2013FUNCTIONS

TABLEX

p-VALUESOBTAINEDBYBONFERRONI–DUNN’S,HOLM’S,ANDHOCHBERG’SPROCEDURESFORTHECOMPAREDDEALGORITHMS

ON30-DIMENSIONALCEC2013FUNCTIONS

TheWilcoxontestandFriedman’stestareemployedtodemonstratethedifferencesbetweenZEPDEandthe?veimprovedDEvariants.TheresultsofstatisticalanalysisareshowninTablesVIIIandIX.TableVIIIshowsthatZEPDEcansigni?cantlyoutperformsjDE,SaDE,JADE,EPSDE,andCoDEon16,19,10,17,and11testfunctions,respectively.Meanwhile,TableIXshowsthattheoverallperformanceoftheproposedalgorithmisthebestamongthesesixDEvariants.Bonferroni–Dunn’s,Holm’s,andHochberg’sproceduresarealsoemployedtodetecttheglobaldifferencesamongthedifferentDEvariants.TableXindicatesthattheaver-ageperformanceofZEPDEissigni?cantlybetterthanthoseofEPSDEandSaDE.AlthoughtheoverallperformanceofZEPDEisnotsigni?cantlybetterthanthoseofjDE,JADE,andCoDE,itcanbeseenfromtheresultsoftheWilcoxontestandFriedman’stestthattheaverageperfor-manceofZEPDEisbetterthanthoseofcomparedDEvariants.

D.ComparisonWithFiveImprovedDEon50-DimensionalProblems(CEC2013)

Inthissection,2850-dimensionalCEC2013testfunctionsareemployedtotesttheperformancesofallcomparedDEalgorithms.Theexperimentalresultsaregiveninsupplemen-tary(TableA4)duetopageslimit.

FromthestatisticalanalysisresultsoftheWilcoxontestpre-sentedinTableXI,itcanbeseenfromTableXIthatZEPDE

226TABLEXI

RESULTSOFWILCOXONTESTON50-DIMENSIONAL

CEC2013FUNCTIONSTABLEXII

RANKINGOBTAINEDBYFRIEDMAN’STESTON50-DIMENSIONAL

CEC2013FUNCTIONS

TABLEXIII

p-VALUESOBTAINEDBYBONFERRONI–DUNN’S,HOLM’S,ANDHOCHBERG’SPROCEDURESFORTHECOMPAREDDEALGORITHMS

ON30-DIMENSIONALCEC2013FUNCTIONS

signi?cantlyperformsbetterthanjDE,SaDE,JADE,EPSDE,andCoDEon16,19,10,19,and12testfunctions.However,theperformanceofZEPDEissigni?cantlyworsethanthoseofjDE,SaDE,JADE,EPSDE,andCoDEonsix,one,seven,three,andsixtestfunctions.

TheresultsofFriedman’stestareshowninTableXII,itcanbeseenfromTableXIIthattheaverageperformanceofZEPDEisthebestamongthesecomparedDEalgorithms.Furthermore,thep-valuesobtainedbyBonferroni–Dunn’s,Holm’s,andHochberg’sproceduresshowninTableXIII.TableXIIIindicatesthattheoverallperformanceofZEPDEissigni?cantlybetterthanthoseofjDE,SaDE,andEPSDEon2850-dimensionalCEC2013testfunctions.Overall,theaver-ageperformanceofZEPDEisthebestamongallcomparedalgorithms.

Theabovefourexperimentalcomparisonsindicatethatfor2530-dimensionaland50-dimensionalCEC2005testfunc-tions,theoverallperformanceofZEPDEissigni?cantlybetterthanthoseofjDE,SaDE,andEPSDEandisbetterthanthoseofJADEandCoDE.However,theexploitationcapabilityofJADEisbetterthanthatofZEPDEbecauseJADEusesagreedymutationstrategy.Theoverallperfor-mance(i.e.,explorationandexploitationcapacities)ofZEPDEisbetterthanthatofCoDE.For2830-dimensionaland50-dimensionalCEC2013testfunctions,exploitationcapac-ityofalgorithmmaybeimportantbecausetheshiftedglobaloptimumofalltestfunctionsarerandomlydistributedin[?80,80]D(searchrangeis[?100,100]D).Theexperimental

IEEETRANSACTIONSONCYBERNETICS,VOL.46,NO.1,JANUARY2016

TABLEXIV

RESULTSOFTHEFRIEDMANANDIMAN–DAVENPORTTEST

WITHDIFFERENTNP

TABLEXV

RANKINGOBTAINEDBYFRIEDMAN’STESTUNDERDIFFERENTNP

TABLEXVI

RESULTSOFKRUSKAL–WALLISTESTUNDERDIFFERENTNP

TABLEXVII

RESULTSOFTHEFRIEDMANANDIMAN–DAVENPORTTESTWITH

DIFFERENTSETP

resultsshowninSectionsV-CandV-Dindicatethattheover-allperformanceofZEPDEissigni?cantlybetterthanthoseofjDE,SaDE,andEPSDE.TheresultsalsoshowthattheexploitationcapabilityofZEPDEisnotsigni?cantlyworsethanthatofJADEandisbetterthanthatofCoDEonmostofthetestfunctions.Overall,theaverageperformanceofZEPDEisbetterthanthoseofthe?vecomparedDEvariants,especiallywhenthetestproblemshavehighdimensions.E.AnalysisofParameterSettings

Inthissection,2530-dimensionalCEC2005testfunctionsareutilizedtoassesstherobustnessoftheproposedalgo-rithmandthreenonparametricstatisticaltests(i.e.,Friedman’stest[46],Iman–Davenport’stest[48],andKruskal–Wallistestwithmultiplecomparisons)areemployedtoanalyzethesensitivityofpopulationsize,Setp,Mspofeachmutationstrat-egyineachgeneration,Bset,sigmacombination,andNZ.ThemaximumnumberofFEsissettobe300000andallobtainedresultsarebasedon30independentrunsforeachcomparedDEvariantinallexperiments.

1)ImpactofPopulationSize:Inthisexperiment,theimpactofpopulationsizeisinvestigatedduetothefactthatappro-priatepopulationsizecanachievegoodbalancebetweenexplorationandexploitationcapabilitiesduringtheentireevo-lutionprocess.SomeoftheparametersettingsarethesameasinSectionV-AexceptforNP,whichissettobe50,75,100,125,and150inthisexperiment.Themeanandstandarddeviationvalueshavebeenprovidedinthesupplementaryattachment(TableA5)duetospacelimitation.

Friedman’s[46]andIman–Davenport’stests[48]areemployedtotestwhethertheperformanceofZEPDEissigni?cantlyaffectedbydifferentpopulationsizesettings.Additionally,theFriedman’stest[46]andKruskal–Wallistest

FANANDYAN:SDEALGORITHMWITHZEPDE227

TABLEXVIII

RANKINGOBTAINEDBYFRIEDMAN’STESTUNDERDIFFERENTSETP

TABLEXIX

RESULTSOFKRUSKAL–WALLISTESTUNDERDIFFERENTSETP

withmultiplecomparisonsareusedtodeterminethesuitablepopulationsizesettingfortheproposedalgorithm.

TableXIVshowsthattheperformanceofZEPDEisnotextremelysensitivetodifferentNP.TablesXVandXVIindi-catethatNP=100isthemostappropriateparametersettingfortheproposedalgorithm.

2)ImpactofSetp:ToinvestigatetheimpactofSetp,whichvariesfrom0to0.4withastepof0.05inthisexperiment.Themeanandstandarddeviationvalueshavebeengiveninthesupplementaryattachment(TableA6)duetospacelimitation.Friedman’s[46]andIman–Davenport’stests[48]areemployedtotestwhethertheperformanceofZEPDEissigni?cantlyaffectedbydifferentsettingsofSetp.Friedman’stest[46]andKruskal–Wallistestwithmultiplecomparisonsareusedto?ndthesuitableparametersettingforthepro-posedalgorithm.TheotherparametersettingsofZEPDEarethesameasinSectionV-A.

TheresultsobtainedbyFriedman’s[46]andIman–Davenport’stests[48]arereportedinTablesXVIIandXVIII.TableXVIIindicatesthattheperformanceofZEPDEissig-ni?cantlyinsensitivetodifferentSetp.Meanwhile,itcanbeseenfromTableXVIIIthattheperformanceofZEPDEisidealwhenSetp=0.2.However,TableXIXshowsthattheperfor-manceofZEPDEissimilarwhenSetpiswithin[0.1–0.2].Basedontheresultsobtainedbytheabovestatisticalanalysis,Setp=0.175,whichisamedianvaluebetween0.15and0.2,issuitablefortheproposedalgorithm.

3)ImpactofMsp:Inthisexperiment,theparame-tersettingsoftheproposedalgorithmarethesameasthoseusedinSectionV-A,exceptforMsp,whichvariesfrom0.01to0.09withastepequalto0.01.Themeanandstandarddeviationvalueshavebeengiveninthesup-plementaryattachment(TableA7)duetospacelimitation.Friedman’stest[46]andIman–Davenport’stest[48]areusedtodemonstratewhethertheperformanceofZEPDEissigni?-cantlysensitivetodifferentMsp.Additionally,theFriedman’stest[46]andKruskal–WallistestwithmultiplecomparisonsareemployedtoidentifythemostsuitableMspinZEPDE.TableXXrevealsnosigni?cantperformancediffer-encesunderdifferentMspintheproposedalgorithm.TablesXXIandXXIIindicatethatMsp=0.01isasuitablechoice.

4)ImpactofBset:Inthisexperiment,todemonstratetheeffectofBset,whichvariesfrom0to1inastepof0.1.TheotherparametersettingsofZEPDEarethe

TABLEXX

RESULTSOFTHEFRIEDMANANDIMAN–DAVENPORTTESTWITH

DIFFERENTMSP

sameasinSectionV-A.Themeanandstandarddeviationvalueshavebeengiveninthesupplementaryattachment(TableA8)duetospacelimitation.Friedman’stest[46]andIman–Davenport’stest[48]areusedtodemonstratewhethertheperformanceofZEPDEissigni?cantlydifferentunderdifferentBset.TableXXIIIshowsthattheperformanceofZEPDEissigni?cantlysensitivetodifferentBset.Friedman’stest[46]andKruskal–WallistestwithmultiplecomparisonsareemployedtoidentifythesuitableBsetinZEPDE.FromTablesXXIVandXXV,itcanbeseenthatBset=0.35(themedianvalueof0.3and0.4)maybeanappropriatechoiceintheproposedalgorithm.

5)ImpactofSigmaCombinations:Inthisexperiment,thein?uenceofthecombinationsofσ1,σ2onZEPDEperfor-manceisinvestigated.TheproposedalgorithmemploysthesameparametersettingsasthoseusedinSectionV-A,exceptforσ1,σ2,whichvariesfrom0to0.8inastepof0.05.Themeanandstandarddeviationvalueshavebeengiveninthesupplementaryattachment(TableA9)duetospacelimita-tion.Friedman’stest[46]andIman–Davenport’stest[48]areusedtodemonstratewhethertheperformanceofZEPDEdif-ferssigni?cantlyindifferentsigmacombinations.TableXXVIindicatesthattheperformanceofZEPDEisnotsigni?-cantlyin?uencedbydifferentsigmacombinations.Friedman’stest[46]andKruskal–Wallistestwithmultiplecomparisonsarealsoemployedtoanalyzethesuitablesigmacombina-tionforZEPDEalgorithm.FromTableXXVII,itcanbeseenthatthecombination(0.5,0.5)providesthebestperformance;however,TableXXVIIIshowsthattheoverallperformanceofcombination(0.35,0.35)isalsoexcellent.Therefore,incon-siderationofthestatisticalanalysisresults(i.e.,Friedman’sandKruskal–Wallistests),thesigmaofZEPDEcouldbeselectedwithintherangeof[0.25,0.55].

6)ImpactoftheNZ:ChangeinNZmayin?uencetheper-formanceofZEPDEbecausedifferentzoningsmayresultindifferentcombinationsofcontrolparameters.ToinvestigatetheeffectofNZ,NZissettobe1,2,4(Fig.1),6,and9.ThezoningimplementationsareshowninFig.4.Intheexperiment,theproposedalgorithmemploysthesameparametersettings

228IEEETRANSACTIONSONCYBERNETICS,VOL.46,NO.1,JANUARY2016

TABLEXXI

RANKINGOBTAINEDBYFRIEDMAN’STESTUNDERDIFFERENTMSP

TABLEXXII

RESULTSOFKRUSKAL–WALLISTESTUNDERDIFFERENTMSP

TABLEXXIII

RESULTSOFTHEFRIEDMANANDIMAN–DAVENPORTTESTWITHDIFFERENTBSET

TABLEXXIV

RANKINGOBTAINEDBYFRIEDMAN’STESTUNDERDIFFERENTBSET

TABLEXXV

RESULTSOFKRUSKAL–WALLISTESTUNDERDIFFERENTBSET

TABLEXXVI

RESULTSOFTHEFRIEDMANANDIMAN–DAVENPORTTESTWITHDIFFERENTSIGMACOMBINATIONS

TABLEXXVII

RANKINGOBTAINEDBYFRIEDMAN’STESTUNDERDIFFERENTSIGMACOMBINATIONS

TABLEXXVIII

RESULTSOFKRUSKAL–WALLISTESTUNDERDIFFERENTSIGMACOMBINATIONS

asthoseusedinSectionV-A,expectforNZ.NZ(a)–(f)denotesixZEPDEvariantsthatusethezoningtypes[Fig.4(a)–(f)].Moreover,1730-dimensionalCEC2005test(i.e.,F1–F17)functionsareused.Themeanandstandarddeviationval-ueshavebeengiveninthesupplementaryattachment(TableA10)duetospacelimitation.Friedman’stest[46]andIman–Davenport’stest[48]areusedtodemonstratewhethertheperformanceofZEPDEissigni?cantlyin?uencedbydif-ferentNZvalues.TableXXIXindicatesthattheperformanceofZEPDEissigni?cantlyaffectedbydifferentNZvaluesandcanbeimprovedbyincreasingNZ.Moreover,fromtheresultsobtainedbytheFriedman’stest[46]andKruskal–WallistestwithmultiplecomparisonsshowninTablesXXXandXXXI.TableXXXshowsthatNZeprovidesthebestperformance.TableXXXIindicatesthatZEPDEisnotsigni?cantlyworse

TABLEXXIX

RESULTSOFTHEFRIEDMANANDIMAN–DAVENPORTTESTWITH

DIFFERENTNZ

TABLEXXX

RANKINGOBTAINEDBYFRIEDMAN’STESTUNDERDIFFERENTNZ

thanotherZEPDEvariantsonanytestfunctions.Therefore,toachievebalancebetweenalgorithmperformanceandruntime,NZ=4isselectedfortheproposedalgorithm.

FANANDYAN:SDEALGORITHMWITHZEPDE229

Fig.4.ZoningtypesofcontrolparametersforNZsis1(a),2(bandc),6(dande),and9(f).

TABLEXXXI

RESULTSOFKRUSKAL–WALLISTESTUNDERDIFFERENTNZ

TABLEXXXII

RESULTSOFTHEFRIEDMANANDIMANIMAN–DAVENPORTTESTSWITH

DIFFERENTZEPDEVARIANTS

Fig.5.

EvolutioncurvesofthemutationstrategiesofZEPDEforF4.

TABLEXXXIII

RANKINGOBTAINEDBYFRIEDMAN’STESTUNDERDIFFERENTZEPDE

VARIANTS

TABLEXXXIV

RESULTSOFKRUSKAL–WALLISTESTUNDERDIFFERENTZEPDE

VARIANTS

Fig.6.

EvolutioncurvesofthemutationstrategiesofZEPDEforF8.

TABLEXXXV

RESULTSOFKRUSKAL–WALLISTESTUNDERDIFFERENTZEPDE

VARIANTSONTHREECEC2005FUNCTIONS

F.Self-adaptivemutationstrategyofZEPDE

Todemonstratetheeffectoftheself-adaptivemutationstrategiesofZEPDE,?veZEPDEvariants(i.e.,ZEPDE1,ZEPDE2,ZEPDE3,ZEPDE4,andZEPDE5usingonlyDE/rand/1,DE/rand/2,DE/best/2,DE/current-to-best/1,orDE/current-to-best/2mutationstrategy)withasinglemutationstrategyareemployedtosolve1730-dimensional

Fig.7.EvolutioncurvesofthemutationstrategiesofZEPDEforF9.

CEC2005testfunctions(i.e.,F1–F17).TheparametersettingsofthesevariantsarethesameasZEPDEinSectionV-A.Themeanandstandarddeviationvalueshavebeengiveninthesupplementaryattachment(TableA11)duetospacelimitation.

230IEEETRANSACTIONSONCYBERNETICS,VOL.46,NO.1,JANUARY2016

Fig.8.EvolutioncurvesoftheweightedaveragevaluesofFandCRinfourregions.(a)EvolutioncurveoftheweightedaveragevaluesofFandCRinregionA.(b)EvolutioncurveoftheweightedaveragevaluesofFandCRinregionB.(c)EvolutioncurveoftheweightedaveragevaluesofFandCRinregionC.(d)EvolutioncurveoftheweightedaveragevaluesofFandCRinregionD.

Friedman’stest[46]andIman–Davenport’stest[48]areemployedtotestwhethertheperformanceofZEPDEissig-ni?cantlyaffectedbydifferentmutationstrategies.ItcanbeseenfromTableXXXIIthattheperformanceoftheproposedalgorithmissigni?cantlyin?uencedbythechoiceofmuta-tionstrategy.Meanwhile,fromtheresultsofstatisticalanalysisobtainedbyFriedman’stest[46]andKruskal–WallistestwithmultiplecomparisonsshowninTablesXXXIIIandXXXIV,itcanbeseenthattheperformanceofZEPDEisbetterthanthoseofZEPDEvariantswithasinglemutationstrategy.Inaddition,tofurtherdemonstratetheeffectivenessoftheself-adaptivemutationstrategiesoftheproposedalgorithm,threetestfunctions(i.e.,F4,F8,andF9selectedfromCEC2005)areusedtotesttheadaptationofmutationstrategies,andthreetypicalevolutioncurvesofmutationstrategiesofZEPDEareshowninFigs.5–7.AsshowninFigs.5–7andTableXXXV,itcanbeobservedthattheevolutionofthemutationstrategiesofZEPDEconformstotheresultspresentedinTableXXXV.G.Self-AdaptiveControlParametersofZEPDE

Inthisexperiment,aCEC2005testfunction(i.e.,F3)isemployedtoanalyzetheself-adaptivepropertiesofcontrolparameters.someoftheparametersettingsofZEPDEarethesameasinSectionV-A.Theevolutioncurvesofthe

G,CRG,FG,weightedaverageofFandCR(i.e.,FWW,BW,A,A

GGGGCRW,B,FW,C,FW,D,andCRW,D)ineachregionareshowninFig.8.Fig.8(a)indicatesthattheweightedaveragevalue

Fig.9.Curvesofthenumberofsigmacombinationsineachregion.(a)CurveofthenumberofsigmacombinationsinregionA.(b)CurveofthenumberofsigmacombinationsinregionB.(c)Curveofthenumberofsigmacom-binationsinregionC.(d)CurveofthenumberofsigmacombinationsinregionD.

ofFgraduallydecreasesduringtheentireevolutionprocess,andtheweightedaveragevalueofCRgraduallyincreaseswhengenerationislessthan1000.Fig.8(b)showsthattheweightedaveragevalueofFgraduallyincreasesthroughouttheentireevolutionprocess,andtheweightedaveragevalueofCRgraduallyincreaseswhengenerationislessthan1000.Fig.8(c)indicatesthattheweightedaveragevalueofFgrad-uallyincreasesduringtheevolutionprocess,andtheweightedaveragevalueofCRgraduallydecreases.Fig.8(d)showsthat

FANANDYAN:SDEALGORITHMWITHZEPDEtheweightedaveragevalueofFgraduallydecreasesthrough-outtheentireevolutionprocessandtheweightedaveragevalueofCR?uctuatesintheevolutionprocess.Overall,itisobservedthatcontrolparametercombinationsofeachzoninggraduallyplayadifferentrolethroughouttheentireevolu-tionprocess.Therefore,ZEPDEcanwellbalancebetweentheexploitationandexplorationcapabilities.H.NumberofControlParameterCombinationsinEachZoning

Toshowthechangesinthenumberofcontrolparametercombinationsineachzoningduringtheevolutionprocess,one30-dimensionalCEC2005testfunction(i.e.,F3)isused.Fig.9showsthatthenumberofcontrolparametercombi-nationscanbeself-adaptivelyadjustedwiththepopulationevolutionineachzoning.

VI.CONCLUSION

Aself-adaptivedifferentialevolutionalgorithmwithZEPDEisproposedinthispaper.TheperformanceofZEPDEiscomparedwiththatof?vefamousDEvariantson2530-dimensionaland50-dimensionalCEC2005testfunctionsand2830-dimensionaland50-dimensionalCEC2013testfunctions.TheexperimentalresultsshowthattheoverallperformanceofZEPDEisbetterthanthoseoftheothercomparedalgorithms,especiallywhenthetestfunctionhashighdimensions.SeveralnonparametricstatisticaltestsarealsoemployedtoevaluatetheperformancesofallcomparedDEalgorithmsandanalyzethereasonabilityofkeyparame-terchoiceofZEPDE.Severalself-adaptivepropertiesofthemutationstrategiesandcontrolparametersoftheproposedalgorithmandthevariationsinthenumberofcontrolparam-etercombinationsineachzoningarediscussedinthispaper.Fromtheexperimentalresults,itisobservedthatthemutationstrategiesandcontrolparametersofZEPDEcanautomati-callyadaptduringtheevolutionprocess,andtheperformanceofZEPDEissigni?cantlyaffectedbytheNZs.Therefore,ZEPDEcanquicklyadapttocomplexandunknownoptimiza-tionenvironmentsandcanmaintainthesearchcapabilitiesofcontrolparametercombinationsofdifferentzoningsduringtheentireevolutionprocess.

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QinqinFanreceivedtheB.S.degreefromWuhanInstituteofTechnology,Wuhan,China,in2007,andtheM.S.degreefromEastChinaUniversityofScienceandTechnology,Shanghai,China,in2011,whereheiscurrentlypursuingthePh.D.degreefromtheDepartmentofAutomation.

Hiscurrentresearchinterestsincludedifferentialevolutionalgorithm,particleswarmoptimization,constrainedoptimization,multiobjectiveoptimiza-tion,andtheirreal-worldapplications.

XuefengYanreceivedthePh.D.degreefromZhejiangUniversity,Hangzhou,China.

HeiscurrentlyaProfessorwithEastChinaUniversityofScienceandTechnology,Shanghai,China.Hiscurrentresearchinterestsincludecomplexchemicalprocessmodeling,opti-mizingandcontrolling,processmonitoring,faultdiagnosis,andintelligentinformationprocessing.