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D:33/10等于0.3Total1010/10等于1Duringreproductioncrossoversoccuratarandomplace(centerofthegenomeforA',B'andC',justafterthefirstgeneforD').Thelinkexistingbetweenthedegreeofadaptationandtheprobabilityofreproductionleadstoatrendtotheriseoftheaveragefitnessofthepopulation.Inourcase,itjumpsfrom7to10.

Duringthefollowingcycleofreproduction,C'andD'willhaveamondescendant:

D':+C':等于

Thenewsubjecthasinheritedtheintendedgenome:hispawshavebeeflippers.

Wecanthenseethattheprincipleofgeicalgorithmsissimple:

Encodingoftheprobleminabinarystring.

Randomgenerationofapopulation.Thisoneincludesageicpoolrepresentingagroupofpossiblesolutions.

Reckoningofafitnessvalueforeachsubject.Itwilldirectlydependonthedistancetotheoptimum.

Selectionofthesubjectsthatwillmateaccordingtotheirshareinthepopulationglobalfitness.

Genomescrossoverandmutations.

Andthenstartagainfrompoint3.

Thefunctioningofageicalgorithmcanalsobedescribedinreferencetogenotype(GTYPE)andphenotype(PTYPE)notions.

SelectpairsofGTYPEaccordingtotheirPTYPEfitness.

Applythegeicoperators(crossover,mutation...)tocreatenewGTYPE.

DevelopGTYPEtogetthePTYPEofanewgenerationandstartagainfrom1.

Crossoveristhebasisofgeicalgorithms,thereisneverthelessotheroperatorslikemutation.Infact,thedesiredsolutionmayhappennottobepresentinsideagivengeicpool,evenalargeone.Mutationsallowtheemergenceofnewgeicconfigurationswhich,bywideningthepoolimprovethechancestofindtheoptimalsolution.Otheroperatorslikeinversionarealsopossible,butwewon'tdealwiththemhere.

D-AdaptationandSelection:thescalingproblem

Wesawbeforethatinageicalgorithm,theprobabilityofreproductiondirectlydependsonthefitnessofeachsubject.Wesimulatethatwaytheadaptivepressureoftheenvironment.

Theuseofthismethodneverthelesssettwotypesofproblems:

A"super-subject"beingtoooftenselectedthewholepopulationtendstoconvergetowardshisgenome.Thediversityofthegeicpoolisthentooreducedtoallowthegeicalgorithmtoprogress.

Withtheprogressionofthegeicalgorithm,thedifferencesbetweenfitnessarereduced.Thebestonesthengetquitethesameselectionprobabilityastheothersandthegeicalgorithmstopsprogressing.

Inordertopalliatetheseproblems,it'spossibletotransformthefitnessvalues.Herearethefourmainmethods:

1-Windowing:Foreachsubject,reduceitsfitnessbythefitnessoftheworsesubject.Thispermitstostrengthenthestrongestsubjectandtoobtainazerobaseddistribution.

2-Exponential:Thismethod,proposedbyS.R.Ladd,consistsintakingthesquarerootsofthefitnessplusone.Thispermitstoreducetheinfluenceofthestrongestsubjects.

3-LinearTransformation:Applyalineartransformationtoeachfitness,i.e.f'等于a.f+b.Thestrongestsubjectsareonceagainreduced.

4-Linearnormalization:Fitnessarelinearized.Forexampleoverapopulationof10subjects,thefirstwillget100,thesecond90,80...Thelastwillget10.Youthenavoidtheconstraintofdirectreckoning.Evenifthedifferencesbetweenthesubjectsareverystrong,orweak,thedifferencebetweenprobabilitiesofreproductiononlydependsontherankingofthesubjects.

Toillustratethesemethods,let'sconsiderapopulationoffoursubjectstochecktheeffectofscaling.Foreachsubject,wegivethefitnessandthecorrespondingselectionprobability.

Subjects1234RoughFitness50/50%25/25%15/15%10/10%Windowing40/66.7%15/25%5/8.3%0/0%Exponential7.14/36.5%5.1/26.1%4.0/20.5%3.32/16.9%Lineartransfo.53.3/44.4%33.3/27.8%20/16.713.3/11.1%Linearnormalization40/40%30/30%20/20%10/10%Windowingeliminatestheweakestsubject-theprobabilityestozero-andstimulatesthestrongestones(thebestonejumpsfrom50%to67%).

Exponentialflattensthedistribution.It'sveryusefulwhenasuper-subjectinducesanexcessivelyfastconvergence.

Lineartransformationplaysslightlythesamerolethanexponential.

Atlast,linearnormalizationisneutraltowardsthedistributionofthefitnessandonlydependsontheranking.Itavoidsaswellsuper-subjectsasatoohomogeneousdistribution.

Conclusion

GeicalgorithmsareoriginalsystemsbasedonthesupposedfunctioningoftheLiving.Themethodisverydifferentfromclassicaloptimizationalgorithms.

Useoftheencodingoftheparameters,nottheparametersthemselves.

Workonapopulationofpoints,notauniqueone.

Usetheonlyvaluesofthefunctiontooptimize,nottheirderivedfunctionorotherauxiliaryknowledge.

Useprobabilistictransitionfunctionnotdeterministones.

It'simportanttounderstandthatthefunctioningofsuchanalgorithmdoesnotguaranteesuccess.Weareinastochasticsystemandageicpoolmaybetoofarfromthesolution,orforexample,atoofastconvergencemayhalttheprocessofevolution.Thesealgorithmsareneverthelessextremelyefficient,andareusedinfieldsasdiverseasstockexchange,productionschedulingorprogrammingofassemblyrobotsintheautomotiveindustry.

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