空间科学与技术学院学术论文集西安电子科技大学Word下载.docx
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空间科学与技术学院学术论文集西安电子科技大学@#@ADecompositionBasedMulti-objectiveParticleSwarmOptimizerforConstraintHandling@#@WeikangNing,BaolongGuo,YunyiYan,JinfuWu,DanZhao@#@(SchoolofAerospaceScience&@#@Technology,XidianUniv.,Xi’an710071,China;@#@)@#@Abstract:
@#@Multi-objectiveparticleswarmoptimization(MOPSO)algorithmsbasedondecompositionhavedrawnalotattentionrecently.DespitethesuccessofdecompositionbasedMOPSO(MOPSO/D)algorithms,itsuseinconstrainedmulti-objectiveoptimizationproblems(CMOPs)remainstobefurtherstudied.MostMOPSO/Dalgorithmsproposedrecentlyaredesignedforunconstrainedproblems.ThusweaimtoextendtheabilityofMOPSO/Dalgorithmsinourstudy.AframeworkofdecompositionbasedMOPSOforconstrainthandling(cMOPSO/D)isfirstproposedinthispaper.ThentwotypicalconstrainthandlingtechniquesarecombinedwithcMOPSO/Dasacomparison,whichresultsintwoversionsofcMOPSO/D:
@#@cMOPSO/Dusingpenaltyfunction(cMOPSO/D-P)andcMOPSO/Dusinglexicographicordering(cMOPSO/D-LO).ThealgorithmsproposedaretestedontenCMOPsandarecomparedwiththreestate-of-the-artalgorithms,cMOEA/D,D2MOPSOandOMOPSO.Experimentalresultssupportedbythestatisticalanalysisofthreequantitativemetrics,togetherwithsometheoreticalanalysis,suggestthattheproposedalgorithmsareeffective,competitiveandpromising.@#@Keywords:
@#@Multi-objectiveoptimization,Decomposition,PSO,Constrainthandling,Penaltyfunction,Lexicographicordering@#@1.Introduction@#@Asaparadigmofevolutionaryalgorithm(EA),particleswarmoptimization(PSO)hasbeensuccessfullyappliedtosolvingmulti-objectiveoptimizationproblems(MOPs)[22,20,11].Intherealworld,however,manyproblemstobesolvedoftenneedtosatisfyseveralequalityand/orinequalityconstraints,andthispresentsanadditionalchallengeformulti-objectiveevolutionaryalgorithms(MOEAs).Solvingconstrainedmulti-objectiveproblems(CMOPs)ideallyoftenrequiresthesearchmakingabalancebetweenthefeasibleandinfeasibleregions,thustheinformationcarriedbytheinfeasiblesolutionscouldbefullyexploited.@#@Manyconstrainthandlingtechniquesareproposedandhavebeenstudiedforalongtimeinsingleobjectiveoptimization[7,13].Penaltyfunctioniswidelyusedforitssimplicity[28,18,33,2].Thefitnessofasolutioniscalculatedbasedonitsobjectivevalueandapenaltyterm.Buthowtodeterminetheamountofpenalizationisratherhard.Toalleviatethisdrawback,someadaptivepenaltyfunctionsarereportedintheliterature[17,8,3,5].Besides,differentconstrainthandlingtechniquesbasedonlexicographicorderingarealsowidelystudied.Deb[9]proposedabinarytournamentselectionoperatortocomparetwosolutions.Basedonthesatisfactionlevelfortheconstraints,TakahamaandSakaiproposedanαconstrainedmethod@#@[29]thatusesαlevelcomparisontocomparetwosolutions.Asanimprovementofαcon-strainedmethod,εconstrainedmethod,whichwasproposedbyTakahamaandSakai[30],adoptsanεlevelcomparisonto@#@comparetwosolutions.AdynamiccontrolofεwasalsoproposedandthecomparisonschemewascombinedwithPSO.Becauseitishardtostriketherightbalancebetweentheobjectiveandthepenaltyterminpenaltyfunctions,RunarssonandYao[23]proposedastatisticrankingmethodinwhichtheconstraintviolationisignoredwithsomeprobability.@#@InrelativelytraditionalParetodominancebasedMOEAs,constraint-dominanceisawidelyusedconstrainthandlingtechnique.Inspiredbythebinarytournamentselectionoperator[9]usedinconstrainedsingleobjectiveoptimization,aconstraint-dominancerelationshipwasproposedbyDeb[10]inNSGA-IItocomparetwosolutions.SimilarconstrainthandlingstrategiesareadoptedbyMOPSO[6]andOMOPSO[27].Besides,penaltyfunctionbasedconstrainthandlingtechniquesarealsoused.Anadaptivepenaltybasedconstrainedhandlingtechniquewasincorporatedwithnon-dominatedsortingbyWoldesenbet[32].Inthismethod,eachobjectivevalueofasolutionismodifiedfirstbasedonitsconstraintviolationandthennon-dominatedsortingisused.@#@Theconceptofdecomposition,whichwasproposedbyZhang[37]recently,hasproventobeefficientinhandlingmanycomplexMOPs.Thefitnessassignmentmechanismadoptedbymulti-objectiveevolutionaryalgorithmbasedondecomposition(MOEA/D)makesiteasierforMOEA/Dtoincorporatemanyconstrainthandlingtechniquesoriginallyinventedforsingleobjectiveoptimization.JanandZhang[14]proposedapenaltyfunctionbasedMOEA/Dforconstrainthandling.Thepenaltyfunctionadoptedusesanadaptivethresholdtocontrolhowheavyaninfeasiblesolutionispunished.Besides,twolexicographicorderingbasedMOEA/Darealsoreported.Asafuddoula[4]introducedanallowableviolationthresholdtodetermineifaninfeasiblesolutionisconsideredasfeasible.Thethre