量子行为粒子群优化算法-英文版

Quantum-behavedParticleSwarmOptimizationOutlineBackgroundQuantumParticleSwarmOptimizationConvergenceoftheParticleExperimentResultsonBenchmarkFunctionsConclusionFutureWorkBackgroundswarmintelligenceoatypeofbiological(social)systemothecollectivebehaviorsofsimpleindividualsinteractingwiththeirenvironmentandeachotheroTherearetwopopularswarminspiredmethodsinswarmintelligenceareas:•AntColonyOptimization(ACO)•ParticleSwarmOptimization(PSO)BackgroundParticleSwarmOptimizationItmimicsthecollectiveintelligentbehaviorof“intelligent”creaturesItwasdevelopedin1995byJamesKennedyandRussellEberhart[Kennedy,J.andEberhart,R.(1995).“ParticleSwarmOptimization”,Proceedingsofthe1995IEEEInternationalConferenceonNeuralNetworks,pp.1942-1948,IEEEPress.]()IthasbeenappliedsuccessfullytoawidevarietyofsearchandoptimizationproblemsInPSO,aswarmofnindividualscommunicateeitherdirectlyorindirectlywithoneanotherineachsearchdirections.Aparticle(individual)iscomposedof:Threevectors:•Thex-vectorrecordsthecurrentposition(location)oftheparticleinthesearchspace•Thep-vectorrecordsthelocationofthebestsolutionfoundsofarbytheparticle•Thev-vectorcontainsagradient(direction)forwhichparticlewilltravelinifundisturbed.Twofitnessvalues:•Thex-fitnessrecordsthefitnessofthex-vector•Thep-fitnessrecordsthefitnessofthep-vector.ParticleSwarmOptimizationTheAnatomyofaParticleTheAnatomyofaParticleIiX=xi0,xi1,…,xin-1P=pi0,pi1,…,pin-1V=vi0,vi1,…,vin-1x_fitness=?p_fitness=?ParticleSwarmOptimizationParticleSwarmOptimizationTheparticlewillmoveaccordingtothefollowingequation:Velocitycalculationvid(t)=w*vid(t-1)+c1*rand()*(pid-xid(t-1))+c2*rand()*(pgd-xid(t-1))Positioncalculationxid(t)=xid(t-1)+vid(t)xid–currentvalueofthedimension“d”ofparticle“i”vid–currentvelocityofthedimension“d”ofparticle“i”.Pid–optimalvalueofthedimension“d”ofparticle“i”sofar.Pgd–currentoptimalvalueofthedimension“d”oftheswarm.c1,c2–accelerationcoefficients.w-inertiaweightfactorParticleSwarmOptimization510152025510152025pbestgbestv(k)v(k+1)510152025510152025PidPgdVid(t)Vid(t-1)ParticleSwarmOptimizationSwarmSearchInPSO,particlesneverdie!Particlescanbeseenassimpleagentsthatflythroughthesearchspaceandrecordthebestsolutionthattheyhavediscovered.Initiallythevaluesofthevelocityvectorsarerandomlygeneratedwiththerange[-Vmax,Vmax]whereVmaxisthemaximumvaluethatcanbeassignedtoanyvid.OncetheparticlecomputesthenewXiitthenevaluatesitsnewlocation.Ifx-fitnessisbetterthanp-fitness,thenPi=Xiandp-fitness=x-fitness.ParticleSwarmOptimizationThealgorithm1.Initialiseparticlesinthesearchspaceatrandom.2.Assignrandominitialvelocitiesforeachparticle.3.Evaluatethefitnessofeachparticleaccordingtoauserdefinedobjectivefunction.4.Calculatethenewvelocitiesforeachparticle.5.Movetheparticles.6.Repeatsteps3to5untilapredefinedstoppingcriterionissatisfied.Quantum-behavedParticleSwarmOptimizationTherearestillsomelimitationsinparticleswarmincludingbutnotlimitedto:ThePSOisnotaglobalconvergence-guaranteedalgorithm.TherelianceofthesearchglobalsearchabilityontheupperlimitofthevelocityreducestherobustofPSOalgorithm.ParameterselectionisanotherproblemQuantum-behavedParticleSwarmOptimizationTheMotivationofQPSOAccordingtothecharacteristicofcollectivenessofswarmintelligence,thepotentialwellofwasbuiltonthepointbetweenpidandpgdTheprobabilitydensityfunctionanddistributionfunctionareWhereLisaparameter.Quantum-behavedParticleSwarmOptimizationTheevolutionequationsoftheQPSOUsingMonteCarlomethod,weobtainthefollowingequationThemeanofthepbestpositionisintroducedLisevaluatedbyTheevolutionequationoftheQPSOTheQPSOAlgorithm(1)Initializepopulation:randomxi(2)do(3)Calculatembestusingequation(10)(4)fori=1topopulationsizeM(5)Iff(xi)f(pi)thenpi=xi(6)pg=min(pi)(7)ford=1todimensionD(8)fi=rand(0,1)(9)P=(fi1*pid+fi2*pgd)/(fi1+fi2)(10)L=a*abs(mbestd-xid)(11)u=rand(0,1)(12)ifrand(0,1)0.5(13)xid=P-L*ln(1/u)else(14)xid=P+L*ln(1/u)(15)UntilterminationcriterionismetQPSOisprovidedwiththefollowingcharacteristics：•EnhancetheglobalsearchabilityofPSOalgorithm•Hasjustonlyoneparameter,easytorealizeandtoselecttheparameter.•ItismorestablethanoriginalPSO.ConvergenceBehavioroftheindividualparticleinQPSOInthestochasticsimulations,pointPisfixedatx=0,andtheinitialpositionoftheparticleissettobe1000,thatisx(0)=1000.ThevalueofContraction-ExpansionCoefficientaissettobe0.7,1.0,1.5,1.7,1.8and2.0respectively,andthenumberofiterationsare1000,1500,5000,1500,50,000,and7000respectively.Thelogarithmicvalueofthedistancebetweencurrentpositionx(t)andpointpisrecorded.ConvergenceBehaviouroftheIndividualParticleinQPSOConvergenceBehavioroftheIndividualParticleinQPSOConvergenceBehavioroftheIndividualParticleinQPSOConvergenceBehavioroftheIndividualParticleinQPSOItcanbeconcludedthatwhena1.7,theparticlewillconvergetothepointP,andwhena1.8,itwilldiverge.Theremustbesuchathresholdvaluea0ininterval(1.7,1.8)thatifaa0,theparticlewillconverge,orelseitwilldiverge(aa0).Wehavetheoreticaldemonstratedusingprobabilistican