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Optimization of Composite Structures by Estimation of Distribution Algorithms

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OPTIMIZATIONOFCOMPOSITESTRUCTURESBYESTIMATIONOFDISTRIBUTIONALGORITHMSByLAURENTGROSSETADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2004

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Copyright2004byLaurentGrosset

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Idedicatethisworktomyparents.

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ACKNOWLEDGMENTSIwanttoexpressmygratitudetoDr.RaphaelT.HaftkaandDr.RodolpheLeRicheforgivingmetheopportunitytocompletethisdualPh.D.Ithankthemforthetimeandeorttheyspenttomakethisjoint-degreeprogrampossible.Ialsothankthemfortheirexcellentscienticguidanceandfortheirnever-endingenthusiasmtoexplorenewareas.Workingwithtwoadvisors,andbeingpartoftwoteamshasbeenagreatsourceofinspirationandanenrichinghumanexperience.IwouldalsoliketothankDrs.Fortunier,Kim,Pardalos,Sankar,Schoenauer,andVautrinforagreeingtobemembersofmyPh.D.supervisorycommitteeandfortakingthetimetoreviewthisdissertation.IamgratefulforthefriendshipofmycolleaguesintheStructuralandMulti-disciplinaryOptimizationResearchGroup,AmitKale,MelihPapila,XueyongQu,PalaniRamu,RalucaRosca,JacoSchutte,SatchiVenkataraman,whomademystud-iesinFloridaapleasurableexperience,andoftheMeMdepartmentintheEcoledesMines,PierreCelle,BernadetteDegache,SylvainDrapier,MarcoGigliotti,JihedJe-didi,Jer^omeMolimard,JoelMonnatte,BenoitSerre,StephaneVacher:theymadelifeinSaint-EtienneascolorfulasinFlorida.Finally,Iwanttothankmyfamilyfortrustingandsupportingmeduringthesefouryears. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. ix LISTOFFIGURES ................................ x LISTOFACRONYMS .............................. xv LISTOFSYMBOLS ................................ xvi ABSTRACT .................................... xviii CHAPTER 1INTRODUCTION .............................. 1 1.1Optimizationofcompositelaminates ................ 3 1.2Fromgeneticalgorithmstoestimationofdistributionalgorithms 7 1.3Goalsofthisresearch ......................... 13 2ESTIMATIONOFDISTRIBUTIONALGORITHMS ........... 15 2.1Preliminary:stochasticmodelofadeterministicfunction ..... 15 2.2Estimationofdistributionalgorithms ................ 18 2.2.1Generalprinciple ....................... 18 2.2.2Illustration ........................... 19 2.3Thegeneralestimationofdistributionalgorithm .......... 20 2.4Selectionschemes ........................... 22 2.5Estimatingthedistributionofpromisingpoints:theoreticalissues 26 2.5.1Theconservationlawforgeneralizationperformance .... 26 2.5.2Simpleexample ........................ 27 2.5.3Thebiasversusvariancecompromise ............ 28 2.5.4Assessingtheaccuracyofanestimate ............ 29 2.6Estimatingthedistributionofpromisingpointsinpractice .... 31 2.7Estimationofdistributionalgorithmsandotherstochasticalgorithms 32 v

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3THEUNIVARIATEMARGINALDISTRIBUTIONALGORITHM ... 35 3.1Algorithm ............................... 35 3.2StudyoftheoriginalUMDA ..................... 37 3.2.1Problemdescription ...................... 38 3.2.2Populationsizeandselectionpressure ............ 39 3.2.3Dimensionality ......................... 46 3.3InvestigationofpossiblemodicationstoUMDA:memoryandmu-tation ................................ 58 3.3.1Memory ............................ 58 3.3.2Mutation ............................ 60 3.3.3Boundontheprobabilities .................. 63 3.3.4Elitism ............................. 64 3.3.5Conclusionoftheparameterstudy .............. 66 3.4Comparisononthreeproblems .................... 66 3.4.1Presentationofthealgorithms ................ 67 3.4.2Constrainedmaximizationoftherstnaturalfrequency .. 67 3.4.3MinimizeA66 .......................... 72 3.4.4Strengthmaximization .................... 74 3.5Conclusion ............................... 76 4THEDOUBLE-DISTRIBUTIONOPTIMIZATIONALGORITHM ... 77 4.1Motivation ............................... 77 4.2Principles ............................... 80 4.2.1Identicationofdependenciesanddatasimplication ... 80 4.2.2Samplingfromtwodistributions ............... 82 4.3Thedouble-distributionoptimizationalgorithm .......... 87 4.3.1Generalalgorithm ....................... 87 4.4Applicationtocompositeoptimization ............... 88 4.4.1DDOAforcomposites ..................... 88 4.4.2Issuesassociatedwiththerepresentationoftheauxiliaryvariabledistribution .................... 91 4.4.32Ddidacticexample ...................... 93 4.4.4Extensionalproblem ...................... 94 4.4.5Extensional-exuralproblem ................. 100 4.4.6Strengthproblem ....................... 104 4.4.7AcomparisonofdiversityinUMDAandDDOA ...... 106 4.5Performancewithoptimizedparameters ............... 115 4.5.1Parameterstudyandbestsetting .............. 115 4.6GeneralizationofDDOAtocontinuousoptimizationproblem ... 119 4.6.1Problemdescription ...................... 119 4.6.2Thealgorithms ......................... 120 4.6.3Resultsanddiscussion ..................... 121 4.6.4Improvementtothealgorithm ................ 125 vi

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4.7Conclusion ............................... 127 5CONCLUSION ................................ 129 5.1Summaryofthendings ....................... 129 5.1.1Incorporationofvariabledependenciesthroughphysics-basedauxiliaryvariables ...................... 130 5.1.2Controlofthediversity .................... 131 5.1.3Combinationofdistribution-basedanddirectionalsearchmech-anisms ............................ 132 5.2Potentialavenuesforfutureresearch ................ 133 6RESUME ................................... 134 6.1Optimisationdestratiescomposites ................ 134 6.2Optimisationstatistique ....................... 135 6.3Contenudelathese .......................... 139 6.4Conclusions .............................. 144 REFERENCES ................................... 146 APPENDICES ACOMPOSITELAMINATEOPTIMIZATION ............... 153 A.1Basicmechanicsoflaminatedplates ................. 153 A.1.1Stinessmatrices ....................... 153 A.1.2Laminationparameters .................... 155 A.2Mechanicalpropertiesusedinthiswork ............... 159 A.2.1Poisson'sratio ......................... 159 A.2.2Coecientofthermalexpansion ............... 160 A.2.3Firstnaturalfrequency .................... 162 A.2.4Strength ............................ 162 BBANDWIDTHSELECTIONFORDDOA ................. 164 B.1Kerneldensityestimation ....................... 164 B.2Determinationofthebandwidth ................... 166 B.2.1Eectofontheaccuracy .................. 166 B.2.2Maximumlikelihoodmethod ................. 169 B.3ApplicationinthecontextofDDOA ................ 169 B.4Eectofthebandwidth:empiricalresults .............. 171 CLAMINATIONPARAMETERPROPORTIONALACCEPTANCE:EX-PERIMENTALRESULTS ........................ 174 vii

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DMULTIOBJECTIVEOPTIMIZATIONOFLAMINATES ........ 176 D.1Problemdescription .......................... 176 D.2Introductiontomultiobjectiveoptimization ............. 178 D.3Geneticalgorithm ........................... 181 D.3.1Laminateencoding ...................... 182 D.3.2Fitnessfunction ........................ 183 D.3.3Geneticoperators ....................... 185 D.3.4Implementation ........................ 188 D.4Results ................................. 188 D.5Concludingremarks .......................... 194 BIOGRAPHICALSKETCH ............................ 196 viii

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LISTOFTABLES Table page 2{1Samplepoints,=20 ........................... 27 3{1Materialpropertiesofgraphite-epoxy .................. 39 3{2Meanbesttnessatthefthgeneration ................. 41 3{3Materialpropertiesofgraphite-epoxy. .................. 68 3{4Materialpropertiesofglass-epoxy. .................... 74 4{1Characteristicsofthethreealgorithms .................. 116 D{1Materialpropertiesofgraphite-epoxyandglass-epoxy ......... 178 D{2Parametersassociatedwiththegeneticoperators ............ 189 D{3Optimumdesignsoftheminimizationofthecompositeobjectivefunction 190 ix

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LISTOFFIGURES Figure page 2{1Stochasticmodelofafunction ...................... 16 2{21Dobjectivefunction ........................... 20 2{3Convergenceofthedistributionpx ................... 20 2{4GeneralEstimationofDistributionAlgorithm ............. 22 2{5Selectionasanoisyclassicationtask .................. 25 2{6Estimatingthedistributionfromasample ............... 27 3{1BasicUnivariateMarginalDistributionAlgorithm ........... 37 3{2FitnesslandscapeA11 ........................... 39 3{3Evolutionofthemeanbesttness .................... 41 3{4Reliabilityoftheoptimizationforvariouscombinationsofpopulationssizesasafunctionofthenumberofiterations ............ 42 3{5Reliabilityoftheoptimizationforvariouscombinationsofpopulationssizesasafunctionofthenumberofevaluations ........... 43 3{6Evolutionoftheprobabilitiesfor1x1,A11maxproblem ...... 45 3{7Orderingconventionforabalancedsymmetriclaminate. ....... 48 3{8Fitnesslandscapeoftherstnaturalfrequencyf1 ........... 48 3{9Expectedconvergencetimeofthestochastichill-climberforthemaxA11andvibrationproblems ...................... 50 3{10Numberofanalysesuntiltheaveragemaximumtnessreaches98%oftheoptimaltness,maxA11problem ................. 54 x

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3{11Numberofanalysesuntiltheaveragemaximumtnessreaches98%oftheoptimaltness,vibrationproblem ................ 55 3{12Evolutionoftheprobabilitydistributions. ................ 56 3{13ComparisonoftheperformanceofSHContheMaxA11problemandthevibrationproblem. ......................... 57 3{14EectofmemoryonUMDA ....................... 59 3{15InuenceofmutationonthereliabilityfortwoUMDAvariants .... 62 3{16Eectofmutationforpm=0:005 .................... 63 3{17Eectofimposingalowerboundontheprobabilitiesonthereliability. 64 3{18Eectofanelitiststrategyonreliability. ................ 65 3{19MeanbesttnessandreliabilityforSHC,UMDA,andGAforthecon-strainedmaximizationoftherstnaturalfrequencyofalaminatedplaten=8 .............................. 70 3{20MeanbesttnessandreliabilityforSHC,UMDA,andGAforthecon-strainedmaximizationoftherstnaturalfrequencyofalaminatedplaten=15 .............................. 71 3{21Evolutionofthemarginaldistributionpx1,UMDA,maxA66problem. 73 3{22ReliabilityandmeanbesttnessofUMDAandSHCforthestrengthproblem ................................. 75 4{1Selectedpointsandunivariatedistributionfortheconstrainedvibra-tionproblem .............................. 78 4{2Interpretationofvariabledependenciesasthejointactionofhiddenvariables ................................. 80 4{3Incorporationofvariabledependenciesthroughauxiliaryvariables. .. 82 4{4Probabilisticacceptancescheme:variancereduction .......... 84 4{5Inuenceofthetwo-distributionschemeonthevariance ........ 86 4{6Flowchartofthedouble-distributionoptimizationalgorithm. ..... 87 xi

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4{7Eectofthetwo-distributionsamplingprocedure ........... 93 4{8FitnesslandscapeofthepenalizedA22forn=2. ............ 95 4{9MeanmaximumtnessandreliabilityfortheconstrainedmaxA22problem,n=6runs. ....................... 98 4{10MeanmaximumtnessandreliabilityfortheconstrainedmaxA22problem,n=120runs. ....................... 99 4{11FitnesslandscapeofthepenalizedCTEproblemforn=2. ...... 101 4{12MeanmaximumtnessandreliabilityfortheconstrainedminCTEproblem,n=6. ............................. 102 4{13MeanmaximumtnessandreliabilityfortheconstrainedminCTEproblem,n=120runs. ....................... 103 4{14Meanbesttnessandreliabilityforthestrengthproblem,n=120runs ................................... 106 4{15DiversityinthecriteriondomainforUMDAandDDOA. ....... 109 4{16DiversityinthevariabledomainforUMDAandDDOA. ....... 110 4{17DistributionofthetnessofthebestsolutionateachiterationforUMDAandDDOA. .......................... 111 4{18EectofmutationforUMDAandDDOA ................ 113 4{19EectofboundsontheprobabilitydistributionspkforUMDAandDDOA .................................. 114 4{20ComparedperformancesoftheoptimizedGA,UMDA,andDDOA. 117 4{21Bestschemesforsevenpopulationsizes,forGA,UMDA,andDDOA. 118 4{22Contoursofthepenalizedtnessfunctionforn=2 .......... 120 4{23EvolutionofthebesttnessforcUMDAandcDDOA,n=2averageover50runs. .............................. 122 4{24ConvergenceoftheprimaryandauxiliarydistributionsforcDDOA,n=2runs ............................. 123 xii

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4{25EvolutionofthebesttnessforcUMDAandcDDOA ......... 124 4{26Eectofaninsucientcoverageofthedesingspace .......... 125 4{27EvolutionofthebesttnessforcUMDAandthemodiedcDDOA .. 126 6{1Principedesalgorithmesaestimationdedistribution ......... 136 6{2AlgorithmeadeuxdistributionsDDOA ................. 142 A{1Ply-numberingconventionforbalancedsymmetriclaminates ..... 157 A{2Feasibledomainforpairsoflaminationparameters .......... 158 A{3Simply-supportedrectangularplate ................... 162 B{1Eectofthebandwidth .......................... 165 B{2AccurateestimateofthedistributionpV1;V3ofselectedpointsfortheconstrainedmaximizationofA22. ................. 167 B{3Inuenceofthebandwidthontheapproximationerrorforasamplesizeof=50points. .......................... 168 B{4Exampleofanapproximation^pV1;V3obtainedfor=50pointsand=0:18. .............................. 168 B{5LoglikelihoodLallasafunctionofforn=6,=30,constrainedmaximizationofA22 .......................... 171 B{6Eectofthetargetpointcreationprocedureonthebesttness ... 172 B{7Inuenceoftheboundariesonthesearchdistribution:reject"andnoreject"targetpointcreationschemes ............... 173 C{1Eectofalaminationparameterprobabilityproportionalacceptanceschemeonthebesttness ....................... 175 D{1Geneticalgorithm ............................. 182 D{2Two-pointcrossover ............................ 186 D{3Simplemutation .............................. 187 D{4Permutation ................................ 187 xiii

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D{5Additionanddeletion ........................... 188 D{6Sandwichstructureoftheoptimumdesign ............... 190 D{7Paretocurvefortheminimumweightandcostdesignofthelaminatedplate ................................... 192 D{8LimitationsoftheweightingmethodforconstructingaconcaveParetofront ................................... 193 xiv

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LISTOFACRONYMSDDOAdouble-distributionoptimizationalgorithmEAevolutionaryalgorithmEDAestimationofdistributionalgorithmESevolutionstratregyGAgeneticalgorithmKDEkerneldensityestimatePBILprobability-basedincrementallearningUMDAunivariatemarginaldistributionalgorithm xv

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LISTOFSYMBOLSOptimizationproblemdenitionx=x1;x2;:::;xndesignvariablesnnumberofdesignvariables,problemdimensionFobjectivefunction,tnessfunctionEstimationofdistributionalgorithmspsxselectionprobabilitypxprobabilitydistributionofselectedpointspopulationsizenumberofselectedpointscandidatepoolsizeselectionratiofortruncationselectionpmmutationprobabilityboundonmarginalprobabilitiesbandwidthofthekerneldensityestimateV=V1;V2;:::;Vmauxiliaryvariablesmnumberofauxiliaryvariables xvi

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Compositelaminates=1;2;:::;nlaminateplyangleswithrespecttoareferencecoordi-natesystem[1=2=:::=n]sbalancedsymmetricstackingsequence:kstandsforthetwo-plystackk=)]TJ/F22 11.955 Tf 12.16 0 Td[(k.The`s'subscriptindicatesthatonlythetophalfofthelaminateisgiven:thelowerhalfisobtainedbysymmetrywithrespecttothemid-plane.V1;V3;W1;W3laminationparameters:contributionofthegeometrytothelaminatestinessEvolutionarycomputationterminolgytnessmeasureofthegoodnessofacandidatesolution=objectivefunctionpopulationsetofcandidatesolutionstotheproblemgenerationiterationchromosomecandidatesolutiongenedesignvariableinthiswork xvii

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMIZATIONOFCOMPOSITESTRUCTURESBYESTIMATIONOFDISTRIBUTIONALGORITHMSByLaurentGrossetDecember2004Chair:RaphaelT.HaftkaMajorDepartment:DepartmentofMechanicalandAerospaceEngineeringThedesignofhighperformancecompositelaminates,suchasthoseusedinaerospacestructures,leadstocomplexcombinatorialoptimizationproblemsthatcan-notbeaddressedbyconventionalmethods.Theseproblemsaretypicallysolvedbystochasticalgorithms,suchasevolutionaryalgorithms.Thisdissertationproposesanewevolutionaryalgorithmforcompositelaminateoptimization,namedDouble-DistributionOptimizationAlgorithmDDOA.DDOAbelongstothefamilyofestimationofdistributionsalgorithmsEDAthatbuildastatisticalmodelofpromisingregionsofthedesignspacebasedonsetsofgoodpoints,anduseittoguidethesearch.Agenericframeworkforintroducingstatisticalvariabledependenciesbymakinguseofthephysicsoftheproblemisproposed.Thealgorithmusestwodistributionssimultaneously:themarginaldistributionsofthedesignvari-ables,complementedbythedistributionofauxiliaryvariables.Thecombinationof xviii

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thetwogeneratescomplexdistributionsatalowcomputationalcost.ThedissertationdemonstratestheeciencyofDDOAforseverallaminateopti-mizationproblemswherethedesignvariablesaretheberanglesandtheauxiliaryvariablesarethelaminationparameters.TheresultsshowthatitsreliabilityinndingtheoptimaisgreaterthanthatofasimpleEDAandofastandardgenetical-gorithm,andthatitsadvantageincreaseswiththeproblemdimension.Acontinuousversionofthealgorithmispresentedandappliedtoaconstrainedquadraticproblem.Finally,amodicationofthealgorithmincorporatingprobabilisticanddirectionalsearchmechanismsisproposed.Thealgorithmexhibitsafasterconvergencetotheoptimumandopensthewayforauniedframeworkforstochasticanddirectionaloptimization. xix

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CHAPTER1INTRODUCTIONToday,compositematerialsareessentialinthedevelopmentofhighperfor-mancestructures.Inparticular,theyhavefoundusageinaerospace,automotive,marine,civil,andsportequipmentapplicationswheretheirhighstiness-to-weightorstrength-to-weight,aswellastheiramenabilitytotailoringaregreatlyappreciated.Howevertheyarecostlyandcarefuldesigniscriticalinordertomakearationaluseofthesematerials:thefreedomthattheygivetothedesignercomesatthepriceofamorecomplexdesignprocessbecausetheboundarybetweenmaterialsandstructuresisblurred.Asaresult,o-the-shelfoptimizationmethodssee PardalosandResende 2002 ,forasurveyofoptimizationmethodsareeithercomputationallytooexpensiveornotapplicabletomanycompositedesignproblems.Thechallengeofoptimizingcompositestructureshasbeentheobjectofintensiveresearchformanyyears,andhasresultedinanumberofeectivetools VenkataramanandHaftka 1999 .Oneofthemostsuccessfuldevelopmentsofthelastdecadeintheeldoflam-inateoptimizationhasbeentheintroductionofgeneticalgorithmsGA.whicharesearchmethodsinspiredbytheinterpretationofnaturalevolutionasanoptimizationprocess.OneofthestrengthsofGAsistheirabilitytohandlenon-lineardiscreteproblems,whichoftenariseincompositelaminateoptimization.Onemayarguethatanotherappealofthesemethodsistheireaseofimplementationanduse.However,whilethemethoditselfissimple,eectiveapplicationtopracticalcasesoftenrequires 1

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2 expertisefromtheuser.Typically,anumberofparametershavetobeadjustedbyhand,basedontheuser'sknowledgeoftheproblemandexperiencewiththealgo-rithm.Inaddition,GAstypicallyrequirealargenumberoffunctionevaluations,whichcanbeprohibitivewhenasingleevaluationtakesminutesorhours.Morerecently,severalattemptshavebeenmadetoreformulategeneticalgorithmsinasta-tisticallymoregeneralform,givingrisetoafamilyofoptimizationmethodscalledestimationofdistributionalgorithmsEDA.Themainbenetofabandoningthebiologicalanalogyistheincreasedcontroloverthesearchitprovides.UnlikeGAs,inwhichpromisingdesignsarecreatedbycombiningfeaturesfromgoodindividualsatrandom,EDAsattempttoexplicitlyidentifythefeaturesthatmaketheseindividualsgood,andtousethisknowledgetocreatenewindividualsthatarelikelytohaveahightness.HoweverthereareonlyfewexamplesofapplicationsofEDAmethodstoengi-neeringproblems,andtoourknowledge,theseadvanceshaveyetnotreachedtheeldoflaminateoptimization.Themainobjectiveofthepresentworkistodevelopecientestimationofdistributionalgorithmsforcompositelaminateoptimization.OurapproachisdictatedbyoneoftheconclusionsoftheNoFreeLunchtheorem WolpertandMacready 1995 ,whichexpressesthefactthatifanalgorithmismoreecientthansomeotheralgorithmononeclassofproblems,thepricetopayislowerperformanceonotherclassesofproblems.Theauthorsformulatethetaskofdevel-opinganewalgorithmasfollows:Theonlyimportantquestionis,HowdoIndgoodsolutionsformygivencostfunctionf?"Theproperanswertothisquestionistostartwiththegivenf,determinecertainsalientfeaturesofit,andthenconstruct

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3 asearchalgorithm,a,specicallytailoredtomatchthosefeatures."Consequently,insteadofcreatingablackboxalgorithmthatworkswellformostproblems,weuseourknowledgeoftheparticularclassofproblemsunderconsiderationtoimproveperformance.Wewillshowthatbyincorporatinginformationaboutthephysicsoftheproblemviacarefullychosenauxiliaryvariables,onecansignicantlyimprovetheeciencyofthesearch.1.1OptimizationofcompositelaminatesThemechanicalpropertiesofcompositematerialscanbetailoredtotheproblemathand.Thisconstitutesanadvantagebutitmakesthedesignofthestructuremorecomplexbecauseitinvolvesnotonlythechoiceofthematerialsandofthegeometry,butalsotheinternalarrangementofthematerial.Laminatedcompositesaremadeupofastackofpliescomposedofstibersorientedalongagivendirectionembeddedinmatrix,suchaspolymerresin.Themacroscopicpropertiesofthelaminatedependonthetypeandthegeometryoftheberscarbon,glass,Kevlar,etc,onthematrixproperties,onthebervolumefrac-tion,onthenumberofpliesandtheirthickness,andontheorientationofthebers.Forunidirectionalcomposites,thestrengthandthestinessareconsiderablyhigherinthedirectionofthebersthaninthetransversedirection.Thepurposeofopti-mizationistodeterminethenumberofplies,theirthicknessaswellastheirmaterialandorientationsoastoextremizecertaincriterion,forinstancethedisplacementatonepoint,theweight,thecost,therstnaturalfrequency,etc.,subjecttoconstraintsonstrengthormanufacturing.Thisisknownasstacking-sequenceoptimization.

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4 Ingeneral,thevariablesinvolvedinlaminatedcompositedesignaremixed,i.e.denedoverdiscreteandcontinuousspaces:somearediscretebydenitionthema-terialhastobechosenfromacatalog,somebecomediscreteduetomanufacturingconstraints.Inparticular,inmanyapplications,theorientationofthepliesisnotarbitrarybutcantakeonlyalimitedsetofdiscretevalues,forexample0,45,and90.Consequentlymanycompositeoptimizationproblemsarecombinatorialprob-lems,notoriousforbeingdiculttosolvebytraditionalgradient-basedmethods.Inaddition,theseproblems,whenexpressedintermsoftheplyanglesasdesignvariables,havebeenshownby Pedersen 1987 topossessseverallocaloptima.Tosummarize,compositelaminateoptimizationproblemsareingeneralmulti-modaldiscreteoptimizationproblems.Overtheyears,extensiveresearchhasbeendevotedtosolvingtheseproblems.Thehighcomputationalcostofavailablediscreteoptimizationalgorithmsledearlyresearcherstorelaxthediscretenessconstraintinordertoapplyreadilyavailablemethodstostacking-sequenceoptimizationproblems.Inadesignforminimumweightunderbucklingconstraints, SchmitandFarshi 1977 forceddiscretenessoftheori-entationsbyusingthetotalthicknessofeachofsomeprespeciedanglesasdesignvariables.Theseweretreatedascontinuousduringtheoptimizationandthenalresultwasroundedtotheclosestmultipleoftheplythicknessavailablefromthemanufacturer.Themethodwasnotcapableofoptimizingthestackingsequence,furthermore,theroundingproceduredidnotguaranteeoptimality.Thedicultyofdealingwithdiscreteplyangleswasaddressedby Shinetal. 1990 whousedagraduallyincreasingpenaltyapproachwhichforcedacontinuoussearchalgorithmto

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5 convergetodiscretevaluesfortheplyorientations.Themaindicultywastochooseanappropriateadaptationstrategyforthepenaltyparameterinordertopreventbothprematureconvergencetoalocaloptimumandconvergencetonon-discretesolutions.Thesecondundesirablecharacteristicsofstacking-sequenceoptimizationproblems,namelythenon-convexitywasremovedby HaftkaandWalsh 1992 forthebucklingloadmaximizationproblem.Theytransformedthenon-linearoptimizationproblemintoalinearprogrammingproblembyformulatingitintermsofply-orientation-identity"designvariables.Theresultingproblemwasthenamenabletoveryecientlinearprogrammingalgorithms.However,themethodislimitedtostinessdesignandcannotaddressstrengthconstraints.Morerecently, Foldageretal. 1998 usedthefactthatmanylaminateoptimizationproblemsareconvexwhenexpressedintermsofhigher-levelmechanicalquantitiescalledlaminationparameters.Theyper-formedacontinuousoptimizationusingply-anglesandlaminationparameterssimul-taneouslywiththeaimofforcingconvexity.Theyusedasequentialprocedure,whereeverimprovingstartingpointswerechosenbasedonlaminationparametersensitivityinformation.Intheearly1990's,stochasticsearchmethodsbegantobeappliedtostacking-sequenceoptimizationbecausetheyexhibitseveraldesirableattributes:rstly,theydonotrequirecontinuityorgradientinformationandcanthereforebeappliedtoin-tegerordiscreteproblems.Secondly,theyareinherentlyglobalandareconsequentlylesslikelytogettrappedinlocaloptima. Lombardietal. 1992 solvedthebuckingloadmaximizationbyusingsimulatedannealing.Thealgorithmdemonstratedhighreliabilityinndingnear-optimaldesigns. CallahanandWeeks 1992 rstapplied

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6 ageneticalgorithmtosolveastacking-sequenceproblembyusingtheplyorienta-tionsasdiscretedesignvariables. LeRicheandHaftka 1993 developedageneticalgorithmthattookadvantageofthephysicsoftheproblembydevisinggeneticoperators,suchasthepermutationoperator,specicallytargetedatmodifyingcer-tainmechanicalquantities,suchasbendingstinesswhileleavingotherquantitiesconstant.Theywereabletoimproveeciencybecausethenumberofunsuccessfulmutationscouldbereduced.Analternatestrategiewasproposedby Autio 2000 toincorporatephysics-basedknowledgeintoatheoptimization:rstacontinuousoptimizationisapplied,withthelaminationparametersasdesignvariables,thenageneticalgorithmisusedtodeterminethestacking-sequencethatyieldstheoptimallaminationparameters.Otherresearchershavereportedsuccessfulapplicationsofstochasticoptimizationapproachestolaminateoptimizationproblems.Forinstance Zabinsky 1998 advocatedtheuseofsimplerandomsearchalgorithmscalledIm-provingHit-and-RunIHRincombinationwithblackbox"numericalsimulationtoolsbecausetheyprovidebothecientandrobustmethods.Duringthelastdecade,geneticalgorithmshavebecomeastandardtoolforstacking-sequenceoptimizatione.g. Punchetal. 1995 ; McMahonetal. 1998 ; Parketal. 2003 .Atthesametime,theeldofgeneticoptimizationhasundergonema-jormutations.Geneticalgorithms,andmoregenerallyevolutionaryalgorithmshavebeenthoroughlystudiedandimproved.Thatresearchhasgivenrisetoanewtypeofgeneticalgorithmsbasedonstatisticalmethods.ThenextsectionexplainshowthetransitionfromstandardGAstosearchdistributionevolutionaryalgorithmscanpotentiallybebenecialtostackingsequenceoptimization.

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7 1.2FromgeneticalgorithmstoestimationofdistributionalgorithmsGeneticalgorithmsGA Holland 1975 ; Goldberg 1989 areinspiredfromthebiologicalmetaphoroftheDarwiniantheoryofevolution,whichstatesthataspeciesiscapableofadaptingtoachangingenvironmentbecausethoseindividualsofthepopulationthatarebestadaptedaremorelikelytohavedescendantsandthereforetopassontheirgoodtraitstothenextgeneration.Theunderlyingassumptionsarethatnaturalselectionproducesindividualsthatareoptimalwithrespecttotheirenvironmentandthatonecanduplicatetheprocessonanarticialoptimizationproblemwithacomputer.Geneticalgorithmsstartbyinitializingapopulationofindividualsatrandom.Eachindividualrepresentsapossiblecongurationofthestructuretooptimize,codedintoastringofnitelength,calledchromosome.Thenevolution"issimulatedbysuccessivelyapplyingselectionofparents,recombination,mutationandreplacementoftheparents,creatingasequenceofpopulationsofincreasinglyt"individualsindividualswithhighobjectivefunctionevaluation.Recombination,orcrossover,isanoperatorthatswapsgeneticmaterialbetweentwoormoreparentindividualstocreatechildren.Mutationintroducesrandomperturbationinanindividual,forinstancebychangingthevalueofonegene.GAsarepartofamoregeneralclassofstochasticalgorithmscalledevolutionaryalgorithms,whichalsoincludesevolutionaryprogrammingEP Fogeletal. 1966 andevolutionstrategiesES Rechenberg 1973 .AllthreealgorithmsarebasedontheDarwinianevolutionprinciple.Althoughinitiallythesemethodshadstrong

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8 dierences,nowadays,thefrontiershaveblurred,andtheydieronlyinimplemen-tationdetails.Mostly,GAstypicallyusebinaryrepresentationandusecrossoverasthemainoperatorandmutationasabackgroundoperator,EPusesrepresentationstailoredtotheproblemanddoesnotusecrossover,andES'sworkmainlywithcon-tinuousrepresentation.WhileEPandESusebothcrossoverandmutation,theyrelyheavilyonmutation,whichisusuallyadaptive.Foramoredetailedintroductiontoevolutionaryalgorithms,thereaderisreferredto Spearsetal. 1993 or Back 1996 .Althoughgeneticalgorithmsusuallydesignateevolutionaryalgorithmsthatusebi-naryrepresentation,wewillusetheexpressioninabroadersensetorefertodiscreteevolutionaryalgorithms,evenwhentheyworkwithnon-binaryvariablesandmakeampleuseofmutation.Sincetheirinception,geneticalgorithmshavebeenthesubjectofintensiveinves-tigationtounderstandhowtheyworkandthetypesofproblemsonwhichtheycanbeexpectedtoperformwellrelativetootheralgorithms. Holland 1975 introducedtheconceptofschemata,orpartiallydenedstringstoanalyzetheperformanceofGAs.ThisledtotheBuildingBlockHypothesis,whichstatesthatGAworkswellwheninstancesoflow-order,shortschemas`buildingblocks'thatconferhight-nesscanberecombinedtoforminstancesofschemasthatconferevenhighertness" Mitchelletal. 1994 .Inotherwords,theeciencyincreaseswhengenesthatworktogethercanbedetectedbypartiallysamplingthedesignspaceandtheirassociationhasahighprobabilityofbeingpreservedduringGAprocessing.AreectiononwaystodetectandpreservelinkagebetweengenesresultedintheemergenceofanewclassofGAs.Initialworkreliedonreorderingoperatortogroupthesegenes,sothatthey

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9 donotgetpartedbycrossover, 1 but GoldbergandBridges 1990 showedthattheseoperatorsweretooslowtobeeective.ThemessygeneticalgorithmmGA, Gold-bergetal. 1989 wasanattempttolearn"thestructureoftheproblembyusingaexiblecodingthatenablesthealgorithmitselftoevolvetheadjacencyrelationbe-tweengenessoastokeepstronglycorrelatedvariablestogether.Byminimizingthedeninglengthofthesebuildingblocks,thechanceofcrossoverbreakingitwouldbeminimized.Thislineofresearchhasledtotheconclusionthatthepowerofnaturalevolutionwasboundedandthatlinkagecouldonlybeecientlylearnedbyexplicitstatisticalmethods Harik 1999 .Attheotherendofthespectrum,theincreasingcomplexityofalgorithmspro-vokedskepticismamongcertainresearchers,suchasBaluja,whoclaimedthatonecoulddispensewiththepopulation Baluja 1994 ; BalujaandCaruana 1995 .Basedonthehypothesisthattheroleofthepopulationistokeepmemoryofgoodregionsofthedesignspace,heproposedtoreplaceitwithanexplicitprobabilityvectorthatsummarizesinformationcollectedfromallthepointsvisitedsofarduringthesearch.Thatvectorisusedtogeneratenewindividualswithhighprobabilityofhavinghightnessfunctionevaluations.Thenewobservationsareinturnusedtoupdatethemodelbymovingtheprobabilitiestowardthebestindividual.Theresult-ingalgorithmiscalledPBIL,forprobability-basedincrementallearning.ExtensivecomparisonswithasimpleGA,whichhasahigherlevelofsophisticationrevealed 1Instandardcrossoveroperators,genesthatlieclosetoeachotherarelesslikelytobeseparatedthangenesthataresituatedfarapartonthechromosome.

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10 thatPBILoutperformsGAonmanystandardtestproblemsseealso Baluja 1993 .Finally,onelastendeavorcontributedtotheemergenceofestimationofdistri-butionalgorithms:theattempttostudyGA'sbehaviorbyMarkovchainanalysistoobtainconvergencetimeestimates NixandVose 1991 .Thesemethodsseeevo-lutionaryalgorithmsasdynamicalsystemsandtrytodeterminetheirasymptoticbehaviorstability,convergencetime,etc.Theyinvolvemodelingthepopulationintermsoftransitionprobabilities.Naturally,researchersofthateldrealizedthatbehaviorssimilartoGA'scouldbeachievedbyreplacingpopulationsbyprobabilitydistributions.Forexample, MuhlenbeinandMahnig 2000 showedthatthetradi-tionalcrossoveroperatorcanbereplacedbyaprobabilisticmodelthatrepresentsthedistributionofgooddesignsundercertainconditions.Allthreeapproachesconvergedtowardtheemergenceofestimationofdistri-butionalgorithms.Thesealgorithmsuseaprobabilisticmodelofgoodindividualsencounteredduringtheoptimizationtoguidethesearchtowardpromisingregionsofthedesignspace.MostoftheresearchconductedintheeldofEDAsisdevotedtotheconstructionofgoodprobabilisticmodels. MuhlenbeinandMahnig 2000 foundthatwithoutselection,thegenefrequen-ciesofapopulationconvergedtoastatewheregenesbecomedecorrelatedfromeachother.Basedonthisconclusion,theycreatedanalgorithmthatexplicitlystorestheunivariateallelefrequenciesintheformofaprobabilityvector.Thealgorithm,calledunivariatemarginaldistributionalgorithmUMDAisverysimilartothePBILalgo-rithm. Hariketal. 1999 developedasimilarprobabilisticalgorithmnamedcompactgeneticalgorithmcGA,whichmostlydiersinitsprobabilityupdaterule.

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11 Allofthesealgorithmsusedsimpleprobabilisticmodelsthatdonotaccountforvariableinteractions.Inotherwords,theyassumethatthevariableshavenocombinedactionindependentvariablesandmonitoronlyunivariatemarginalfre-quencies.Inmanypracticalsituations,however,variablesdoinuenceoneanother.Forinstance,intheknapsackproblem, 2 thesizeoftheitemsalreadychosenaectsthechoiceoftheremainingitems,orinstackingsequencedesignformaximumbuck-lingload,somegroupsofpliescandeterminethebucklingmode,andalterthemostdesirableangleforotherplies.Thereforeitisimportanttoaccountforinteractionbetweenvariables.Apracticalwayofmodelingvariableinteractionintheprobabilis-ticsenseistouseBayesiannetworks.Bayesiannetworksaremadeupofastructuredirectedacyclicgraph,whichindicatesthevariablesthatareinterdependent,andnumbersconditionalprobabilities,whichexpressthestrengthoftherelationship.Althoughintheoryitispossibletomodelanyorderofinteractionbetweenvariables,thenumberofmodelparameterstheconditionalprobabilitiesassociatedwitheachlinkofthegraphtoestimatefromthedata,andhencethesizeofthesamplerequiredtoobtainanaccurategraphincreasesveryrapidlywiththeorderoftheinteractionsconsidered.Inaddition,havingahighordermodelmaybeharmful,generatingonlyduplicatesofthepopulationbeingmodeled:thisisespeciallydetrimentalwhentheregionsalreadyvisiteddonotcontaintheoptimum,becausethemodelwillprevent 2Taskofselectingitemstobepackedintoaknapsack.Theitemsarechosenfromalargenumberofobjectswithdierentvolumes.Onecanchooseasmanyitemsofwhatevervolumetogointotheknapsackasonelikes,providedthatthetotalvolumeoftheknapsackislled.

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12 theexplorationofnewareas,aswillbeseeninSection 4.6 .Thesimplesttypeofinteractionsarepairwiseinteractions,whichrelatetwovari-ablesgenes.Severalapproacheshavebeenproposedtoincorporatepairwiseinter-actionsintotheprobabilisticmodel,dependingonwhatmeasureofcouplingisused,andhowthemodelisconstructed.Eventhoughingeneralanarbitraryjointdis-tributioncannotbeexpressedintermsofpairwiseconditionalprobabilities,therstorderinteractionmodelcanbettedtothedistributioninthesenseofsomedistance. DeBonetetal. 1997 usedachainmodeltorepresentthejointdistributionofpromis-ingindividuals.Theiralgorithm,calledMIMICformutualinformationmaximizinginputclusteringlooksforthegenepermutationandtheconditionalprobabilitiesthatminimizetheKullback-Lieberdistancewithrespecttothetruedistributionoftheselectedindividuals.TheyappliedMIMICalgorithmtosolveseveraldeceptiveprob-lemsproblemswhereunivariatefrequenciesalonedrivetheoptimizationawayfromtheoptimumandobservedconsiderablegainoverGAandPBIL. BalujaandDavies 1997 extendedMIMICalgorithmbyusingdependency-treeinsteadofchainmodel,thusoeringmoreexibilityintherepresentationofjointdistributions.EvenhigherorderinteractionscanberepresentedbyusingBayesianNetworks. Pelikanetal. 1999 developedanalgorithm,calledBayesianOptimizationAlgorithmBOAthatcanmodelanyorderofinteractionbetweenvariables.Experimentalresultsonsev-eraltestproblemsindicatethatBOAbenetsfromlearningsecondorderinteractionsonfunctionsinvolvingstrongvariableinteractions,whilesimpleGAsorzerothorderprobabilisticalgorithmsaremoreeectiveonproblemsthatdonotexhibitstrongdependencies.

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13 Todate,therehavebeenonlyfewapplicationsofEDAstoengineeringproblems.TheonlyapplicationthatwefoundwasacomparisonoftherelativeperformancesofGAandPBILfortwocombinatorialproblems:aturbinebalancingproblemandastockcuttingproblem Carter 1997 .CarterfoundthatGAperformedsignicantlybetterthanPBILonbothproblems.Onepossibleexplanationforthebetterperfor-manceofGAonthatproblemmaybethatiteectivelyusesaswapoperator,whichexchangestwogenesatrandom,whichcannotbeobtainedwiththePBILalgorithm.1.3GoalsofthisresearchThepresentworkaimsatdevelopingecientstrategiesforstackingsequenceop-timizationthroughtheimplementationandimprovementofestimationofdistributionalgorithms.WestartbyimplementingsimpleEDAstosolvelaminateoptimizationproblems,withonlyminorchangestotheoriginalUMDAalgorithm.Thenmodica-tionsareproposedtoimprovetheperformance.Inparticular,themainemphasisofthisworkistheincorporationofphysics-basedknowledgetothealgorithmtoimprovetheaccuracyofthestatisticalmodelofpromisingregions,andhencetheeciencyofthesearch.Thisdissertationisorganizedasfollows: inChapter 2 ,ageneralintroductiontoEDAsisprovided.Generalprinciplesarepresented,alongwiththeinherentdicultiesassociatedwiththeestimationofdistributions; Chapter 3 presentsasimpleEDAimplementationanditsapplicationtocom-positelaminateoptimization.Thealgorithm'sperformanceisinvestigated,andimprovementsareproposed.

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14 inChapter 4 ,physics-basedknowledgeisusedtoimprovetheperformanceofthealgorithmintroducedinChapter 3 .Thisisachievedbymonitoringtheprobabilitydistributionsofgoodindividualsbothintheplyanglespaceandinthelaminationparameterspacetoobtaincomplementaryinformationaboutpromisingregions.Thegoalistomodelvariableinteractionswithtwosimplestatisticaldistributions. Chapter 5 providesasummaryofthedissertationandidentiesfutureareasofresearch. Chapter 6 isasummaryofthisworkinFrench,incompliancewiththejointdegreeagreementbetweentheUniversityofFloridaandtheEcoledesMinesdeSaint-Etienne.

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CHAPTER2ESTIMATIONOFDISTRIBUTIONALGORITHMSInthischapter,ageneralintroductiontoestimationofdistributionalgorithmsEDAsisprovided.Estimationofdistributionalgorithmssearchthedesignspacebyiterativelyestimatingthedistributionofpromisingpointsandusingittoguidethesearchinsubsequentgenerations.OnegoalofthischapteristoapproachEDAsfromastatisticalstandpointinordertoidentifydicultiesandoutlinepotentialavenuestoimprovetheireciency.AfterintroducingaconceptualEDAdealingwithprobabilitydistributionsonly,theactualalgorithmisobtainedbyestimatingthesedistributionsfromsamplesofpoints.2.1Preliminary:stochasticmodelofadeterministicfunctionLetFxbeadeterministicfunctiontobemaximizedoveradomainD.ThegoalofglobaloptimizationistondthesetofmaximaO=fx2D:8y2D;FyFxgsee HorstandPardalos 1995 ,forasurveyofglobaloptimizationmethods.WhenpriorinformationabouttheformofFisavailableforinstanceifweknowthatFislinear,orconvexoverD,thenitsvalueatsomepointsofthedesignspacecanbeextrapolatedtootherpointsofD,andinformationaboutthelocationoftheoptimumcanbeextracted.However,whennopriorinformationabouttheformoftheobjectivefunctionisavailable,assumptionsneedtobemade.AconvenientwaytoexpressuncertaintyaboutthetrueformofFistodeneapriordistribution 15

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16 overasetoffunctions, 1 andupdatethatdistributiononthebasisofincomingobservations,therebyreducingtheuncertaintyaboutF.EvenifFisdeterministic,itwillberepresentedasaprobabilitydistributionreectingourlackofknowledge,asillustratedconceptuallyinFigure 2{1 aFunctionscompatiblewiththeobserva-tions. bContoursofpFjx.ThecombinationofapriorprobabilityabouttheformofFxandtheobservationsyieldsadistributionsoverD.Figure2{1:RepresentationofouruncertaintyabouttheobjectivefunctionasaprobabilitydistributionpFjx. ThisrepresentationofuncertaintiesabouttheobjectivefunctionisexplicitlyusedinglobaloptimizationalgorithmscalledstochasticoptimizationSOalgorithms 2 seeforexample Betro 1990 ; Schoen 1990 ,and TornandZilinskas 1987 ,Chap.6forageneralintroduction.GivenastochasticmodelofF,SOdeneheuristicstochoosethenextpointtobeevaluatedsoastoextremizesomecriterion.Twocommon 1Often,thispriordistributionisonlyimplicit,intheformofabiastowardsimple,smoothfunctions.2Sometimes,thesealgorithmsarereferredtoasBayesianAlgorithms,becausetheyuseBayes'ruletoupdateprobabilitieswhennewobservationsaremade.

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17 criteriaare: 1. maximizetheprobabilitythatFexceedssomevalueF0; 2. maximizetheexpectedvalueofF.EstimationofdistributionalgorithmsEDAsderivefromthisstochasticrepre-sentationoftheobjectivefunction.Theyfocusonthesetofgoodpoints"SforexamplethesetofpointswhosetnessexceedsF0.ThestochasticmodelpFjx,constructedfromasetofobservationsxi2S,i=1;:::;N,combinedwiththedef-initionofthesetofpromisingpointsSdeterminestheprobabilityofFbeinginS,denotedbypF2Sjx.TheprobabilitydistributionpxjF2SisthenobtainedbyBayes'rule:pxjF2S=pF2Sjxpx pF2S;.1wherepF2Sisanormalizingconstant,andthepriorprobabilitypxreectsapotentialbiastowardcertainregionsofthespace.Whennoregionispreferredapriori,theprobabilitydensityofpromisingregionspxjF2SisequaltotheprobabilitythatthevalueofFbelongstoS.OuruncertaintyabouttheformofFgivesrisetotheprobabilisticmodelofpromisingregions,whichdictatesaprobabilisticsearchstrategyimplementedines-timationofdistributionalgorithms.TheideaofEDAsistodirectlymodelthesetofpromisingpointsSunlikeSO,whichmodelsthefunctionFandsamplefromittoobtainmoregoodpoints,andeventuallyndtheoptimum.ThegeneralprinciplesofEDAswillbeprovidedinthenextsections.

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18 2.2Estimationofdistributionalgorithms2.2.1GeneralprincipleLetFxbeatnessfunctiontobemaximizedoveradomainD.LetOdesignatethesetofoptimaO=fx2D:Fx=Foptg,whereFoptisthemaximumifFoverD.Theprincipleofestimationofdistributionalgorithmsistoconstructasequenceofdistributionsfptxgt2NthatconvergestowardtheuniformdistributionoverthesetofglobaloptimaO.Thesequenceofdistributionsisobtainedasfollows: 1. p0xisuniformoverD 2. pt+1xisthedistributionoverDobtainedafterapplicationofatness-basedselectionoperatortoallthepointsofthedomain.Theselectionoperatorisdenedbyitsprobabilitypsx2[0;1],whichistheprobabilityofacceptingapointxinanacceptance/rejectionprocess.UsingBayes'rule,therecurrenceequationreadspt+1x=ptxpsx ps;.2whereps=RDptxpsxdxisanormalizingconstant.ForthesequencetoconvergetotheuniformdistributionoverO,twoconditionsarerequired: 1. Fxi=Fxjpsxi=psxjtheselectionprobabilitiesoftwopointshavingthesametnessfunctioneval-uationarethesame, 2. letDt=fx2D:ptx>0g,letRtF=fFx:x2Dtg,

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19 8t>0;9Ft02RtFsuchthatFxiFt0Fxj>Ft09>>=>>;psxi
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20 wherecanbechosenarbitrarily.Theprobabilitydistributionevolvesaccordingtopt+1x=pxjS=pSjxpx R10pSjxpxdx=fxptx R10fxptxdx:.4Thedistributionattimest=0,t=5,t=10,andt=20isshowninFigure 2{3 .Startingfromauniformdistributionover[0;1],ptxgraduallyfocusesonhigh-tnessregions,anddegeneratesintoaDiracfunctioncenteredaboutthehereuniqueglobaloptimum. Figure2{2:The1Dobjectivefunctionhasthreelocalmaximaatx=0:11,x=0:61,andx=1:00,butasingleglobalmaximumatx=0:11. Figure2{3:Convergenceofthedistri-butionpxtowardtheglobaloptimumx=0:11 2.3ThegeneralestimationofdistributionalgorithmEstimationofdistributionalgorithmsconstructasequenceofprobabilitydis-tributionsfptxgt2Nthatconvergestoauniformdistributionoverthesetofglobaloptima.However,explicitlycalculatingthedistributionsusingformulassuchasEq. 2.2 isnotreasonable,asisimpliescomputingthetnessfunctioneverywhere,which

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21 rendersthewholeoptimizationuseless.Inpractice,thedistributionspt+1xarenotcalculatedexplicitly,butapproxi-matedfromanitesampleofsizesimilartotheparentpopulation"inthecontextofevolutionarycomputation,obtainedbytness-basedselectionamongasetofpointsthechildpopulation"inEAsgeneratedfromptx.Sincetheselectionprobabilitydistributionpsxissimulatedfromanitesam-ple,sothatthesecondconditionofSection 2.2.1 maynotbesatised,causingse-lectionerror,andthedistributionptxofpromisingpointsattimetisestimatedbasedonanitenumberofpoints,whichgeneratesestimationerrorthiswillbeaddressedindetailinthenextsection,thepracticalalgorithmcanonlyachieveanapproximatedistribution^ptx,whichisnotidenticaltothetheoreticaldistributionptxingeneral.Theactualtrajectoryf^ptxgt2Ninthespaceofprobabilitieswillbedierentfromthetheoreticaltrajectory.Dependingonthequalityoftheapprox-imationsemployed,itmayormaynotconvergetotheoptimum.Thehat"aswellasthesubscripttwilloftenbedroppedintheremainderofthisworkforsimplicity.ThegeneralowchartofEDAsispresentedinFigure 2{4 .Thealgorithmiscomprisedofthreemainsteps,besidestheinitialization,wherethedistributionp0xischosenwhennotpriorknowledgeaboutthelocationoftheoptimaisavailable,auniformdistributionoverDisused.Therststepconsistsingeneratingapopulationofpointsbysamplingfromptx.Then,tness-basedselectionisappliedtoobtaingood"points.Thenewdistributionpt+1xisnallyestimatedfromtheselectedpopulation.Thevaluesofthepopulationsizeandtheselectedpopulationsizedeterminetheaccuracyoftheapproximationspsxandpt+1x.Theprocedureis

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22 Figure2{4:GeneralEstimationofDistributionAlgorithm repeateduntilastoppingcriterionusuallyaxednumberoffunctionevaluationsismet.Theperformanceofestimationofdistributionalgorithmsisdeterminedbytwofactors: thechoiceoftheselectionschemeprobabilitydistributionpsx,whichdeter-minesthebiastowardtterpointsofthepopulation; thechoiceoftheprobabilitydistributionmodel,andmethodadoptedtoesti-matethedistributionofselectedpoints.Thenextsectionsaredevotedtothesetwocriticalcomponentsofthealgorithm.2.4SelectionschemesInthetheoreticalEDApresentedinSection 2.2.1 ,thetransitionfromthedis-tributionptxofpromisingpointsattimettothedistributionattimet+1is

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23 accomplishedbyapplyingselectiontothewholesearchspaceD.Intheactualal-gorithm,thegoalofselectionistogenerategood"pointsthatwillbeusedtoestimatethedistributionofpromisingpoints.Aselectionschemeischaracterizedby,thenumberofpointssampledfromptx,andtheselectionprobabilitypsxi,i=1;:::;.ThevalueofdeterminesthedeparturefromthetheoreticalselectionprobabilitytherstconditionofSection 2.3 requiringthattheselectionprobabilityofhigher-tnesspointsbehigherthanthatoflower-tnesspointsmaynotbesatised:largervaluesofwilllowertheselectionerror.Whenistoosmall,high-tnessareasmaybeignored,andthealgorithmmayfailtolocatetheglobaloptima,aswillbeobservedinSection 4.6 .Selectiondeterminesthebiastowardtterpointsofthepopulation.LetP=fa1;a2;:::;agdesignatethepopulationofindividualssortedindecreasingorderoftheirtnessFai+1Fai.Manydierentselectionschemescanbedevised:tnessproportionalselection,tournamentselectionkpointsaredrawnfromthepopulation,andthepointhavingthehighesttnessevaluationiskept,Bolzmannselectionpsx/Fx,etc.Inthiswork,twoschemeswillbeused: Truncationselection: givenaninitialpopulationofindividuals,the=bc,2[0;1],individualswiththehighesttnessevaluationarechosen.Theselec-tionprobabilitydistributionwithoutreplacementis:psai=8>>><>>>:1 if1i0otherwise

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24 Linearrankingselection: theprobabilityofselectinganelementaiofthepopu-lationisproportionaltoitsrankiinthepopulation.Theselectionprobabilitydistributionwithreplacementis:psai=2)]TJ/F22 11.955 Tf 11.955 0 Td[(i+1 +1Thechoiceofaparticularselectionschemedeterminestheselectivepressure"alsocalledselectionintensity,orbiastowardtterpointsofthedistribution:strongselectivepressuremeansthatonlytheverybestpointswillappearintheselectedpopulation.Severalmeasuresoftheselectivepressurehavebeenproposed: GoldbergandDeb 1991 introducedthetakeovertime,ornumberofgenerationsuntilthebestindividualllsupthewholepopulationunderthesoleactionofselectionnomutationandnocrossover, Muhlenbein 1998 borrowedtheselectionintensityfromtheeldofbreeding, BlickleandThiele 1996 usedtheconceptoflossofdiversityd,whichisdenedastheproportionofindividualsthatdisappearintheprocessofselection.Thelossofdiversityofvariousschemeswasquantiedby WieczorekandCzech 2002 :theyobtainedvaluesofd=0:711fortruncationselectionofratio=0:3,andd0:432forlinearrankingselectionthemaximumlossofdiversityisobtainedforaninnitepopulation;smallerpopulationsachievealowerselectivepressure.Theoptimalselectionschemeisproblem-dependent:itistheresultofatrade-obetweenfastconvergencetohigh-tnessregions,andtheriskofprematureconver-gencetolocaloptimawhennitepopulationsareused.Indeed,selectionisanoisyclassicationtask:consideringtwocompetingcandidatepointsx1andx2,thegoal

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25 istoretaintheonewhichismoreinformativeaboutpromisingregions.Practically,givenapartition 3 ofx=xI;xII,suchthatFx=FIxI+FIIxIIsuchpar-titionsaresometimescalledschema",forma" Radclie 1992 orhyperplane"intheEAliterature,andonewantstoidentifygoodvaluesofxIbasedontheobservedresponseF,thedecisionwillbeinuencedbythevalueofthepartitionxII.TheobservedtnessFxIoftheoptimalvaluexIofthepartitionxIisinfactarandomvariableFxI=FIxI+FIIxII,wherexIIcantakearbitraryvalues.Thisgivesrisetoanoisydecisiontask,asshowninFigure 2{5 :thegraphschematicallyshowsthedistributionpFjgoodoftheobservedtnessoftheoptimalvalueFxI,andthedistributionpFjbadofsomenon-optimalcompetingvalue. Figure2{5:Selectionisanoisyclassicationtask,astheinuenceofothervariablesaectsthetnessofasubregion. Clearly,thedecisiontochooseonecandidateovertheotherwillbedictatedby 3Forinstance,xI=x1;x2;x3andxII=x4:::;xn.

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26 theintrinsicgoodnessofthepartitionunderconsiderationmeansofthedistribu-tions,butalsobytheinuenceofothervariables,whichdeterminesthevarianceofthedistributions MillerandGoldberg 1995 .Thebestselectionstrategydirectlydependsonthespecictnessconguration:whentheresponseisdominatedbyonepartition,strongselectionwillmasktheeectofotherpartitions,hencegoodvaluesofthesemightbelost.Inthosesituations,weakerselection,whichtakesintoaccountinformationconveyedbypoorercandidatesisappropriate.Ifallpartitionsinuencethetnessequally,strongerselectioncanbeused.2.5Estimatingthedistributionofpromisingpoints:theoreticalissues2.5.1TheconservationlawforgeneralizationperformanceTheestimationofadistributionisacommontaskinmachinelearning,inpar-ticularinpatternclassication,andahostoftechniqueshavebeendeveloppedtoaddressit.Anumberoftheoretical,aswellaspracticalissuesassociatedwiththistaskarewell-known,anddiscussedatlengthintheliterature.Oneclassicalprob-lemconsistsinchosingagoodmodeltoapproximateadistributionfromasample.Supposewewanttoestimateadistributionpx,x2D,onthebasisofsamplepointsxi,i=1;:::;distributedaccordingtopx.Afundamentalresultisthat,intheabsenceofanyproblem-dependentinformation,thereisnosinglebestapprox-imatedistribution^pxthatdescribesthesampledistribution.ThisresultsfromtheNoFreeLunchNFLTheoremsalsocalledConservationLawforGeneralizationPerformancethatprovethatgeneralizationfromanitesampleofpointscannotbeaccomplishedunlessadditionalinformationaboutthedistributionisavailable

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27 Wolpert 1992 ; Schaer 1994 .ItwillbeimportanttobeawareofthesetheoreticallimitationswhenconstructingapproximationstothedistributionofpromisingpointsandoftheimplicitassumptionsmadetoovercometheNFL.2.5.2SimpleexampleToillustratethedicultyofestimatingadistributionfromasampleofpoints,letusconsiderthesampleof=20pointsgiveninTable 2{1 .Thesepointsxi2[0;1],i=1;:::;weregeneratedaccordingtoanunknowndistributionpx,whichweseektoestimate. Table2{1:Samplepoints,=20 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 0.81 0.20 0.23 0.85 0.79 0.34 0.31 0.77 0.24 0.34 0.51 0.38 0.66 0.22 0.79 0.48 0.38 0.70 0.55 0.63 Figure 2{6 showsthesamplepoints,andthreedierentpossibleestimateddis-tributions^px.Althoughthethreedistributionsareverydierentfromoneanother, Figure2{6:Estimatingthedistributionfromasample.Intheabsenceofaprioriknowledgeaboutthedistribution,onecannotdiscriminatebetweenthethreecandi-dateapproximations.

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28 allthreeareequallyvaliddescriptionsofthesample: thedistributionp1xhastwomainmodes,andseverallocalmaxima; thedistributionp2xhasonlytwomodes; thedistributionp3xconsidersthatallthepointsweregeneratedfromauni-modaldistribution,andthattheapparentlocaldensitymaximaresultfromvariabilityinthesample.Todiscriminatebetweenallthepossibleapproximationstothetruedistributionpx,onehastomakeassumptionsaboutitsstructure,orincorporatepriorinforma-tionaboutthetargetfunction.2.5.3ThebiasversusvariancecompromiseLetpxbethetargetdistributiontobeestimated,^pxjDtheapproximationtopxobtainedbasedonthedataD=fx1;:::;xg.Sinceallthesamplepointsareinstancesofpx,therewillberandomvariationsinD,and^pxjDwillreectthesevariations.Toobtainameaningfulestimateoftheerrorincurredwhenusingaparticularmodel,oneneedstoaverageoverallpossiblesamples.Theusualmeasureoftheerrormadebytheapproximationatpointxistheroot-mean-squarepredictionerror:erms=q ED^pxjD)]TJ/F22 11.955 Tf 11.955 0 Td[(px2;.5whereEDdesignatestheexpectedvaluewithrespecttosamplevariability.Thistotalerrorcanbebrokendownintotwocomponentsasfollows:e2rms=ED[^pxjD)]TJ/F22 11.955 Tf 11.955 0 Td[(px]2| {z }bias2+ED^pxjD)]TJ/F22 11.955 Tf 11.956 0 Td[(ED[^pxjD]2| {z }variance:.6

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29 Therstcomponent,theexpectedvalueofthedierencebetweentheapproximationandthetruevalueiscalledbias;thesecondcomponent,whichdescribesthemagni-tudeofthevariationsabouttheapproximation'sexpectedvalueiscalledvariance.Bothtermscontributetothetotalerror.Biasistypicallyhighwhentoosimpleamodelisusedtoapproximatethedistribution,itisalsocalledmodelingerror.Vari-anceresultsfromvariabilityinthesamplepoints.Itisalsoreferredtoaslearningerrorandcanbereducedbyincreasingthesamplesize.Thedicultyofestimatingadistributionfromsamplepointsliesinthetrade-obetweenbiasandvariance,assophisticatedmodelswillbemoreexible,andhavealowbias,butbeextremelysensitivetosamplevariability.Incontrast,thevarianceachievedbyasimplemodelwillbelower,butatthecostofalargerbias.2.5.4AssessingtheaccuracyofanestimateEventhoughtheoreticallythereexistsnoabsolutecriterionfordecidingwhichoftwoapproximationsbestdescribesasetofsamplepointswhennoaprioriinfor-mationaboutthetargetdistributionpxisavailableNFLtheorem,inpractice,heuristicmethodshavebeenproposedtochoosebetweencompetingmodels,andtoestimatethemodelparameters.Thosemethodsimplicitlymakeregularityassump-tionsaboutthedistribution Schaer 1993 :infact,theideaofndinganoptimumoveradomainwithoutexhaustiveenumerationrestsontheassumptionthatthet-nessfunctionpossessessomeregularityattributesthatallowustoextrapolatefrompreviousobservations,hencewerestrictsourattentiontoparticularclassesoffunc-tions,aspointedoutby Raoetal. 1995

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30 GivenasetD=fx1;x2;:::;xgofindependentidenticallydistributeddatapointsandasetofcompetingmodelsM=fm1;m2;:::;mkg,threeclassesofmethodscanbeusedtochoosethemodel^mthatbestapproximatesthedata Dudaetal. 2000 ,Chapter9: MaximumlikelihoodMLmethods choosethemodelwhichmaximizesthelike-lihoodofthedata:^m=maxi=1;:::;kpDjmi=maxi=1;:::;kYj=1pxjjmi:Amajordisadvantageofthesemethodsistheirtendencytoovertthedata.AvariantofMLmethodsisthemaximumaposterioriMAPmethod,wheresomemodelscanbefavoredoverothersbyassigningthemahigheraprioriprobabilitypmi.TheaposterioriprobabilityofamodelafterthedatahasbeenobservedisobtainedbypmijD=pDjmipmi pD;wherepDisanormalizingfactorthatisusuallyignored.Themodelthatmaximizestheaposterioriprobabilityisretained. MinimumdescriptionlengthmethodsMLD recastthebias-variancecompro-miseintoamoregeneralproblemofndingthemostcompactrepresentationofadataset.TheinformationtheoryconceptofKolmogorovcomplexityKtheminimumnumberofbitsrequiredtorepresentdataonacomputerisusedtomeasurethetotalcomplexityofthemodelmiandthedescriptionofthedata

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31 setinmi:Kmi;D=Kmi+KDusingmi:Themodelthatachievesthebesttrade-ominimumKmi;DbetweenthemodelcomplexityKmiandthemodel'sabilitytoecientlyrepresentthedataKDusingmiisretained. Cross-validationmethodsCV areclassicalheuristicstrategiesusedinstatis-ticstoestimatethegeneralizationaccuracyofamodel.TheyconsistinsplittingthedataDintoksetsDiofequalsize=k.Thenkapproximationsarecon-structedbyleavingoutonesetDiandestimatingpxbasedonk)]TJ/F15 11.955 Tf 10.412 0 Td[(1remainingsets.Eachtime,adierenterroriisobtained,andthegeneralizationerrorisestimatedasthemeanofallthei's.Acommonvariantofcross-validationmethodsistheleave-one-outmethod,whichisobtainedfork=.Inthiswork,themaximumlikelihoodmethodandthecross-validationmethodwillbeusedMLisinvokedimplicitlyinSection 3 toestimatethemarginalproba-bilities,andCVisusedinSection 4 tochoosethewindowsizeofthekerneldensityestimate.2.6EstimatingthedistributionofpromisingpointsinpracticeIfthereisnoaprioricriterionforchoosingthebestmodel,howdowegoaboutconstructinganapproximatedistribution^ptx?Basedontheaboveconsiderations,ourapproachwillbedictatedbytwogeneralrules: SimplerisbetterOccam'srazor: Eventhoughthereisnoproblem-independent

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32 theoreticalreasonforpreferringsimplermodelsoversophisticatedones,aprin-cipleofteninvokedinmachinelearningisOccam'srazoroverttingavoidance,whichrecommendsthat,foragiventotalerror,modelsinvolvingfewerparam-etersbefavored. Friedman 1997 showedforclassicationtasksthatthebiaserrorofsimplemodelscanbecompensatedbylowvarianceerror,therebymak-ingthemmoreaccuratethanmoresophisticatedmodelsEDAscanbeviewedasclassiersthatdiscriminatebetweengoodandbadregionsofthesearchspace. Incorporateasmuchinformationabouttheproblemaspossible: Anotherwayofreducingerrorconsistsinincorporatingasmuchinformationaboutthetargetdistributionaspossible.2.7EstimationofdistributionalgorithmsandotherstochasticalgorithmsEstimationofdistributionalgorithmscanbeviewedasageneralizationofevolu-tionaryalgorithms Back 1996 ,andassuch,theybearmanysimilaritieswithgeneticalgorithms Goldberg 1989 andevolutionstrategies Rechenberg 1973 .Theycanalsobelikenedtoamorerecentfamilyofstochasticoptimizationalgorithms,calledMarkovchainMonteCarlo GelfandandSmith 1990 EvolutionstategiesES searchfortheoptimumbyselectinggoodpointstheparentsoutofapoolofcandidatepoints;creatingnewpointsthechildrenbyapplyingperturbationsusuallyGaussianmutationtotheselectedpoints.Ifthepopulationofselectedpointsisconsideredasawhole,theprocessofchoosingaparentandapplyingaGaussianmutationisequivalenttosamplingfromthedistributionofparents,estimatedbyavaryingwindowsizekernel

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33 methodseeAppendix B .ThespecicityofESliesinthestrategyimplementedtoselectthelocalbandwidthi.e.themutationstandarddeviationbyanauto-adaptivescheme,i.e.thegoodnessofaparticularvalueofthebandwidthisjudgedbasedonthethetnessofthepointsthataregeneratedusingthatvalue. GeneticalgorithmsGA proceedbyselectinggoodpointstheparentsoutofapoolofcandidatepoints;creatingnewpointsthechildrenmainlybyapplyingrecombinationoperatorscrossovertotheselectedpoints.Mutationisappliedtoasmallportionofthechildren.Viewedfromastatisticalstandpoint,thespeciccrossoveroperatorimplementeddenesanimplicitstatisticalmodelfortheestimationofpxbydeterminingthemarginalfrequenciesconsideredforthecreationofnewpoints.Forexample,usingauniformcrossovertocreatechildrenamountstosamplingfromunivariatemarginalfrequenciesoftheparents. MarkovChainMonteCarloMCMC algorithmssimulatesamplingfromaprob-abilitydistributionpxbyconstructingaMarkovchainxt;t2Nwhoseinvariantdistributionispx.MCMCuseatwo-stepproceduretogeneratenewpoints:asetofcandidatepointsiscreatedbasedonthecurrentstatextandaproposaldistributionqx;thenthenewpointsareacceptedorrejectedaccordingtoanacceptanceprobabilityAxnewjxcurrent.TheparallelbetweenconventionalEDAsandMCMChasbeendrawngivingrisetohybridalgo-rithmscalledeMCMC,forevolutionaryMCMC DruganandThierens 2003 :

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34 EDAscanbeviewedasnon-homogeneousMCMCswhosestatesarepopula-tionsofpoints,withproposaldistributionattimetequaltothedistributionofpromisingpointsptx,andacceptanceprobabilitycommonlysetto1,eventhoughotherreplacementstrategiesarepossible.Theideaisthatafterasu-cientnumberofiterations,MCMCwillsampleuniformlyfromthesetofglobaloptima.

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CHAPTER3THEUNIVARIATEMARGINALDISTRIBUTIONALGORITHMInthischapter,asimpleestimationofdistributionalgorithm,calledUnivariateMarginalDistributionAlgorithm,ispresented.Itisbasedontheassumptionthatthedesignvariablesarestatisticallyindependent.Thebasicalgorithmistestedonlam-inateoptimizationproblems,andtheinuenceofseveralparametersisinvestigated.Thedistribution-basedsearchmechanismiscomparedtoapoint-perturbation-basedsearchmechanism.Thenimprovementstothealgorithmareproposed,andthere-sultingalgorithmisappliedtomorerealisticlaminateoptimizationproblems.3.1AlgorithmLetFxbeafunctionoveradomainD.Ourobjectiveistondtheoptimax.AsintroducedinChapter 2 ,estimationofdistributionalgorithmssearchthedesignspacebyestimatingthedistributionofpromisingregions.Thedistributionofselectedpointsatagiventimeoftheoptimizationdependsontheformoftheobjectivefunction,andontheselectionprocedure.Inparticular,ithasbeenshown Muhlenbeinetal. 1999 thatwhentheobjectivefunctionisasumofcontributionsfromsinglevariables, 1 andBolzmannselectionisusedpsx/expFx,whereisaparameterthatdeterminestheselectivepressurethedistributionofselected 1SuchfunctionsarecommonlyreferredtoasadditivelydecomposablefunctionsADF. 35

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36 pointstakesthesimpleformpx1;x2;:::;xn=nYk=1pxk;.1wherepxkistheunivariatemarginaldistributionofvariablexk.Inotherwords,thevariablesarestatisticallyindependent,andthecompletedistributioncanbeob-tainedfromtheunivariatemarginaldistributionsonly.Thisgreatlysimpliestheprobabilityestimationprocess.Thealgorithmresultingfromthevariableindepen-denceassumption,calledUnivariateMarginalDistributionAlgorithm MuhlenbeinandMahnig 2000 ,ispresentedinFigure 3{1 .Thealgorithmisidenticaltothegen-eralEDA,exceptthatthegeneralformofpxisreplacedbytheunivariatemodel,thusconsiderablydecreasingthenumberofparameterstobeestimated,whichsim-pliesthealgorithm,reducesthecomputationaloverheadrequiredtoestimatethedistribution,anddecreasestheuncertaintyontheparametersduetovariabilityintheselectedpoints.InthisrstvariantofUMDA,norandomexplorationcomponentsuchasmutationisimplemented,becauseourgoalistoobservedthepurestatisticalsearchmechanisms.ThepotentialbenetsofsuchoperatorswillbeinvestigatedinSection 3.3 .Inthiswork,wedealwithdiscretevariablesxktheberorientationofeachply,whichcantakecvaluesai;i=1;:::;c,hencetheprobabilityfunctionspxkarediscreteprobabilityfunctions.Tospecifypxk,weneedtoknowtheprobabilityofeachvalueai,pki=pxk=ai.Sincethedistributionofeachvariableintheselectedpointsfollowsamultinomialdistributionofprobabilitypxk,themaximumlikelihoodvaluesofpkiaresimplygivenbythefrequenciesfkiofeachvalueaiinthe

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37 Figure3{1:BasicUnivariateMarginalDistributionAlgorithm.Thegeneralprobabil-itydistributionofselectedpointsisreplacedbyaproductofunivariatedistributions. populationofselectedpoints.Byincorporatinginformationabouttheformofthedistribution,onecanobtainamoreaccurateestimateofpx.However,thereisapricetopayforthesimplicationachieved:ifthevariableindependenceassumptionisnotsatised,theestimateddistributionwillbeerroneous,andtheresultingalgorithmmayfailtolocatetheoptima.Inthefollowingsections,weshallapplyUMDAtoADF,andstudytheirbehavioronmorecomplexfunctions.3.2StudyoftheoriginalUMDAUMDA,asanyEDA,theoreticallyworkswithprobabilitydistributions,inprac-ticethedistributionsareestimatedfromnitesamplesofselectedpoints.Thevalueofdeterminestheaccuracyoftheestimatedpki:thelargerthepopulationsize,thelowerthestandarderrorontheseestimatedmodelparameters.Thechoiceof

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38 thepopulationsizewillthereforewillbedrivenbytwoconsiderations:theaccuracyofpxkandthecomputationalcostoneiswillingtopayforit.TheotherfactoraectingUMDA'sbehavioristheselectivepressure.Thepresentsectioninvestigatestheinuenceofthesetwofactorsonthealgorithm'sperformance.3.2.1ProblemdescriptionWerststudiedtheconvergencepropertiesofUMDAforaproblemthatsatisesthevariableindependenceassumption.ThegoalistounderstandthestrengthsandlimitationsofUMDAandmoregenerallyofEDAinanidealsetting.Weconsideredtheproblemofmaximizingthelongitudinalin-planestinessA11ofabalancedsymmetricgraphite-epoxylaminate 2 [x1=x2=:::=xn]s:maximizeA11=h"U1+nXk=1U2cos2xk+U3cos4xk#;.2wherethetotallaminatethicknesswash=0:2in,andU1,U2,andU3aremate-rialinvariantsseeAppendix A formoredetailsaboutthemechanicsofcompositelaminates.Thematerialpropertiesofgraphite-epoxyaregiveninTable 3{1 .Thedesignvariablesaretheberanglesxk,k=1;:::;n,wherexkdesignatestheangleofthekthplyinthelaminatewithrespecttoareferencecoordinatesystem.Forthisproblem,theberanglesweretobechosenfromf0;15;30;45;60;75;90g,andthenumberof-stacksinahalf-laminatewassetton=1040pliesintotal. 2Alaminateissymmetricifthestackingsequenceissymmetricwithrespecttothemid-plane,andbalancedif,foreach-degreeply,thereisa)]TJ/F22 11.955 Tf 9.299 0 Td[(-degreeplyinthelaminate.

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39 Theobjectivefunctionhadonlyonemaximum,at[020]sallthebersarealignedwiththelongitudinalaxis.Thelongitudinalin-planestinessofthatlaminatewasA11=4:38106lb/in.ItisapparentfromEquation 3.2 thattheobjectivefuncTable3{1:Materialpropertiesofgraphite-epoxy E12:18107psiE21:38106psiG121:55105psi120:26 tionbelongstotheclassofADFs,henceitmakessensetousetheunivariatemodelgiveninEquation 3.1 toapproximatethedistributionofA11-basedselectedpoints.Figure 3{2 showsthetnesslandscapeforn=2. Figure3{2:FitnesslandscapeA11forn=2x1andx2areindegrees. 3.2.2PopulationsizeandselectionpressureInthebasicformofUMDA,onlytwofactorsgovernthealgorithm'sbehavior:theselectionscheme,whichdeterminespsxtheselectionschemeencompassesthe

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40 selectionmethodandtheselectivepressureandthenumberofpointsusedtoestimatethedistributionofpromisingpointsptx.Thepurposeofthissectionistoinvestigatetheirrespectiveinuenceonthealgorithm'sperformance.UMDAwasappliedtotheA11maximizationproblem.Fivedierentselectedpopulationsizes2f10;20;50;100;200gweretried,andtruncationselectionofratios2f0:1;0:3;0:5gcf.Section 2.4 wereused 3 .Asanyglobaloptimizationalgorithm,anidealestimationofdistributionalgo-rithmshouldpossesstwooftenconictingattributes,asemphasizedby Back 1996 ,Ch.4inthecontextofevolutionaryalgorithms:highconvergencevelocityandhighconvergencereliability.Inthisstudy,theoptimizationreliabilityRwasestimatedastheproportionofrunsthatndtheoptimuminaxednumberofevaluations.Themeanbesttnessateachiterationwasusedasameasureoftheconvergencevelocity.Bothcriteriawereestimatedbasedon50independentruns,sothatthestandarderrorofRwase=p R)]TJ/F23 7.97 Tf 6.586 0 Td[(R p 50Rfollowsabinomialdistribution,whichismaximumat0.07forR=0:5.Figure 3{3 showstheevolutionofthemeanbesttnessduringtheoptimizationforthe15combinationsofpopulationsizeandselectionratiothemeanbesttnessatthefthgenerationisgiveninTable 3{2 tohelptheinterpretation.Thegraphclearly 3Thischoiceofasanindependentparameterasdependentparameter,deter-minedbythevalueofisunusual:inevolutionarycomputation,theusualapproachistostudytheinuenceofthepopulationsize.Therationaleforthischoiceistodistinguishfactorsthatinuencethenominalvalueofthesearchdistributionthebiasandfactorsthataecttheaccuracyoftheestimatethevariance.

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41 showsthreefamiliesofcurves,correspondingtotheselectionratio:thealgorithmconvergesfastertohighfunctionevaluationregionsforstrongselection.Thisisnotsurprising,asallthevariableshavethesameweightinthetnessfunction,thereforehigh-tnesspointsarealwaysmoreinformativeaboutgoodregionsthanlower-tnesspoints.Withinafamilyofcurvesidenticalselectionratio,theconvergencevelocity Figure3{3:Evolutionofthemeanbesttnessasafunctionofthenumberofitera-tions.Theconvergencevelocityincreaseswhenselectionbecomesstronger. increasesmonotonicallywiththesamplesize,aslargesamplesyieldmoreaccurateestimatesofthedistributionofgoodpointspx. Table3{2:Meanbesttnessatthefthgeneration 0.10.30.5 10 4.334.144.0520 4.364.284.0750 4.384.334.19100 4.384.364.24200 N/A4.374.28

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42 Theoptimizationreliabilitydependsbothontheselectionratioandontheselectedpopulationsize,asshowninFigure 3{4 .Foragivenvalueof,theaccuracyofthedistributionpxisdirectlyrelatedtothesamplesize:thelargerthevalueof,thesmallerthevarianceofthemarginalfrequenciesfkioftheselectedpoints.Thiseectisevenmoresignicantforlargevaluesofweakselection.Whenweakselectionisused,theexpectedvalueoftheproportionofpointsthatcontaintheoptimalvaluexkisbarelyhigherthanthatofnonoptimalvalues.Toguarantythattheestimatedvalueoftheproportionishigher,onehastoreducethevariance,henceincreasetheselectedpopulationsize. Figure3{4:Reliabilityoftheoptimizationforvariouscombinationsofpopulationssizesand,asafunctionofthenumberofiterations.Thenalreliabilityincreaseswiththeselectedpopulationsize. Whiletheevolutionofthereliabilityasafunctionofthenumberofiterationsallowsustounderstandhowthealgorithmparametersaecttheaccuracyofthedistributionpx,hencethesearchmechanism,thenumberoffunctionevaluations

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43 isamoremeaningfulmeasureofthecostassociatedwithanoptimization.Figure 3{5 showsthereliabilityofthe15algorithmvariantsagainstthenumberofanalyses.Usingthatnewscale,anewhierarchybetweenthetestedschemesappears:forsmall Figure3{5:Reliabilityoftheoptimizationforvariouscombinationsofpopulationssizesand,asafunctionofthenumberoffunctionevaluations. selectedpopulationsizes=10and=20,thebestreliabilityisobtainedforstrongselection 4 ,whileforlargeselectedpopulations=100and=200,thereverseorderisobserved.Thiscanbeexplainedbythefactthatthevarianceoftheproportionofgoodpointsintheselectedpopulationislargeforsmallnumbersof 4Notethattheclassicalresultthathighselectionratesrequirelargepopulations ThierensandGoldberg 1993 considerastheindependentparameter,consequentlyincreasingtheselectivepressureresultsinlargersamplingerrorsintheselectedpointsandinapoorerperformance.Whenthenumberofselectedpointsistheindepen-dentparameter,theselectionpressurecanbevariedwithoutaectingthesamplingerror,explainingthedierentconclusionspresentedhere.

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44 selectedpoints,consequentlyaselectiveschemehastobeusedtohelpdiscriminate.Incontrast,whenlargenumbersofselectedpointsareused,thevarianceissucientlysmalltoallowagooddiscrimination,sothatthenumberofevaluationsateachiterationdeterminestheeciency.ThisresultshowsthetwoendsofthespectruminEDAs:smallpopulationsleadtostatisticalerrors,whereasalgorithmsusinglargepopulationssuerfromhighcomputationalcosts.Thiscanbeeasilyunderstoodbyexaminingtheevolutionoftheprobabilitieswithtime.Figure 3{6 showstheevolutionoftheprobabilitydistributionp1,start-ingfromauniformdistribution,duringtherstteniterations 5 .Thecombinedeectsofselectedpopulationsizeandselectionratioareclearlyvisibleonthegraphs:forthesameselectionintensity,thedistributionconvergesmoresmoothlytowardtheoptimumvalue1=0Figuresaandbwithincreasing.Withasmallselectedpopulationof10points,thedistributionlosesvariablevalues:forinstancevariablevalues60degreesand90degreesvanishbetweenthersttwoiterations.Inthiscase,thelossofvaluesspeedsuptheconvergencebyfavoringoptimalvalues,however,pre-matureconvergenceoftheprobabilitiescanleadtothelossofoptimalvalues,whichpreventsthealgorithmfromndingtheoptimum,sincenodiversityinjectionmech-anismsuchasmutationoralowerboundonthemarginalfrequenciesisprovided.AlargerselectedpopulationenablesUMDAtoobtainmoreaccurateestimatesofalltheprobabilities,resultinginamorereliableoptimizationscheme.Foragiven 5Sinceallvariablesareinterchangeable,theevolutionofothervariableswouldexhibitsimilartrends.

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45 a=100,=10 b=1000,=100 c=200,=100 Figure3{6:EvolutionoftheprobabilitiesfortheA11maxproblem.

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46 selectedpopulationsize=100,Figuresbandcshowthefasterconvergenceofthedistributionwhenstrongerselectionisused:p1convergesinonly6iterationsfor=0:1,whileithasnotfullyconvergedafter10iterationsfor=0:5.Tosummarize,theselectionratiodeterminestheconvergencespeedofthealgo-rithmfromonegenerationtothenext.Foridenticalselectionratios,largeselectedpopulationsprovidemoreaccuratestatisticsaboutgoodalleles,therebypreventingprematureconvergence.However,ifalargeselectionpressureassociatedwithlargeselectedpopulationsconstitutesthebestcongurationforfastconvergenceintermsofiterations,theincreasednumberofanalysesrequiredtorenetheevaluationofprobabilitiesmayresultinanoverallschemewhichislessecientthanalgorithmsbasedonsmallerpopulations.Thisproblemcomesdowntotheissueoftheallocationofresourcesbetweenexplorationandexploitation:usingasmallerselectedpopula-tionresultsininaccurateparameterestimates,andcanpotentiallydrivethesearchtowardpoorindividuals,butsincefeweranalysesareperformedateachgeneration,thelossinaccuracyiscompensatedbytheincreasednumberofiterationspermitted,givenaxedbudget.3.2.3DimensionalityAnimportantcharacteristicofoptimizationalgorithmistheimpactofanin-creaseofthenumberofvariablesonthetimetoconvergence,whichiscalledthealgorithmtimecomplexity.Inthissection,wecompareUMDA'stimecomplexitytothatofarandomsearchalgorithmcalledStochasticHill-ClimberSHC.SHCisoneofthesimpleststochasticalgorithms,andhasbeenproposedasabaselinemethod

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47 forevaluatingevolutionaryalgorithms'performance JuelsandWattenberg 1995 .Thealgorithmsareappliedtotwoproblems:themaximizationofthein-planelongi-tudinalstinessintroducedinSection 3.2.1 ,andthemaximizationoftherstnaturalfrequencyofasimplysupportedrectangulargraphite-epoxylaminatedplate.Problemdescription:frequencymaximizationTherstnaturalfrequencyofasimplysupportedrectangularplateispropor-tionaltothesquarerootofthefollowingexpression:f1=D11 L4+2D12+2D66 L2W2+D22 W4.3whereL=20inandW=15inarethelengthandwidthoftheplate,theDijtermsarethecoecientsofthebendingstinessmatrixcf.Appendix A :D11=U1h3 12+4 3U2nXk=1tkz2k)]TJ/F22 11.955 Tf 11.956 0 Td[(t2kcos2xk+4 3U3nXk=1tkz2k)]TJ/F22 11.955 Tf 11.955 0 Td[(t2kcos4xk.4D22=U1h3 12)]TJ/F15 11.955 Tf 13.151 8.088 Td[(4 3U2nXk=1tkz2k)]TJ/F22 11.955 Tf 11.955 0 Td[(t2kcos2xk+4 3U3nXk=1tkz2k)]TJ/F22 11.955 Tf 11.955 0 Td[(t2kcos4xk.5D66=U5h3 12)]TJ/F15 11.955 Tf 13.151 8.088 Td[(4 3U3nXk=1tkz2k)]TJ/F22 11.955 Tf 11.955 0 Td[(t2kcos4xk.6D12=U4h3 12)]TJ/F15 11.955 Tf 13.151 8.088 Td[(4 3U3nXk=1tkz2k)]TJ/F22 11.955 Tf 11.955 0 Td[(t2kcos4xk.7wheretheUtermsarethematerialconstantsgiveninTable 3{1 ,theplythicknesstkwasxedat0.005in,andzkreferstothepositionofthekthplyinthelaminate,asillustratedinFigure 3{7 .LiketheA11maximizationproblem,thisproblemisanADF,thereforeUMDAtheoreticallyconvergestotheoptimumxi=60,i=1;n.However,contrarytothe

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48 Figure3{7:Orderingconventionforabalancedsymmetriclaminate. rstproblem,thez'sintheDij'sgiveahierarchicalstructuretotheproblembecauseplieslocatedinouterlayershavemoreweightthanthoselocatedininnerlayers,ascanbeseeninFigure 3{8 casen=2.Asaresult,thedistributionscorrespondingtotheoutermostpliesareexpectedtoconvergefasterthanthosecorrespondingtoinnerplies.ThefunctionsA11andf1areunimodal,sothatSHCwillalsoyieldxforthetwoproblems. Figure3{8:Fitnesslandscapeoftherstnaturalfrequencyf1forn=2.Thevariablex1istheorientationoftheouterplies,andx2istheorientationofthecoreplies.

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49 AcompetitorforUMDA:stochastichill-climberSHCsearchesthespacebychoosinganinitialpointx=x1;x2;:::;xnatran-domandapplyingrandomperturbationstoit:thealgorithmchangesthevalueofonevariablechosenatrandomtoanadjacentvalueandacceptsthenewpointonlyifitimprovesthetnessfunction.Theprocedureisstoppedwhenaxednumberoffunctionevaluationshasbeenperformed.TheoreticalanalysisofSHCThefollowinganalysisconsidersastochastichillclimberSHCoperatingonaunimodalfunction.IftheSHCisatapointwherekoutofthenvariablesarecorrectlyset,theexpectedtimebeforeoneofthenonoptimalvariableisperturbedisn=n)]TJ/F22 11.955 Tf 10.214 0 Td[(k.Therandomperturbationcanthentakethevariableclosertotheoptimumornot,withprobabilities1=2neglectingdistortionsduetolimitsonthevalues.Theexpectedtimeforonebenecialstepisthen2n=n)]TJ/F22 11.955 Tf 12.25 0 Td[(k.Letdidenotetheaveragedistancebetweenthei-thvariablesofarandompointandtheoptimum,di=1 ccXj=1jxji)]TJ/F22 11.955 Tf 11.955 0 Td[(xij;.8wherexjiisthejthpossiblevalueoftheithvariable.Inthecasespresentedhere,thevalueofallthevariablesattheoptimumisthesamexi=0formaxA11andxi=45forthevibrationproblem,thereforetheaveragedistancetotheoptimumisthesameforallvariablesi,di=dbutitvarieswiththeproblem.Foreachvariablethatisnotcorrectlyset,anaverageofdstepsintherightdirectionisneededtoreachtheoptimum.Bysumminguptheexpectedtimesofeachbenecialstep,oneobtainstheexpectedtimetolocatetheoptimumfromarandomstartingpointthathask

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50 optimalvariablesTk=n)]TJ/F21 7.97 Tf 6.587 0 Td[(1Xi=k2dn n)]TJ/F22 11.955 Tf 11.955 0 Td[(i:.9Tkcannowbeaveragedoverallrandomstartingpoints,whichyieldstheexpectedconvergencetimeofanSHConaunimodalfunction,TSHC=1 2nnXk=0CknTk=nd 2n)]TJ/F21 7.97 Tf 6.587 0 Td[(1nXk=0Cknn)]TJ/F21 7.97 Tf 6.586 0 Td[(1Xi=k1 n)]TJ/F22 11.955 Tf 11.955 0 Td[(i:.10Itcanbeshown Garnieretal. 1999 thatEquation 3.10 yieldsaconvergencetimeoforderOnlnnforlargen.EstimatedfromEq. 3.10 andmeasuredconvergencetimesaverageover50independentrunsarecomparedforboththemaxA11andthevibrationproblemsinFigure 3{9 .Thequalitativepredictionofthedependencyofthenumberoffunctionevaluationsintermsofthedimensionniscorrect.BecausedissmallerinthevibrationproblemthaninthemaxA11problem,convergenceisfasterintheformercase,whichisalsocorrectlypredicted. Figure3{9:Expectedconvergencetimeofthestochastichill-climberforthemaxA11andvibrationproblems.SHCndstheoptimuminOnlnnevaluations.

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51 TheoreticalanalysisofUMDAAunivariatemarginaldistributionalgorithmisnowconsidered.In Muhlenbeinetal. 1999 ,thebehaviorofaUMDAwithtruncationselectionwasstudiedfortwotestfunctions,OnemaxandInt,thathavecommonfeatureswiththemaxA11andthevibrationproblems.IntheOnemaxproblem,thenumberofone'sinabinarystringismaximized.AsinmaxA11,thefunctionisseparable,andeachvariablehasthesamecontributiontotheobjectivefunction.TheIntfunction,Int=nXi=12i)]TJ/F21 7.97 Tf 6.586 0 Td[(1xi;.11isalsomaximizedonbinarystrings.Asinthevibrationproblem,thefunctionisseparableandthereisagradualinuenceofthevariablesonthefunction.Inthevibrationproblemhowever,thedierenceinsensitivityoftheobjectivefunctiontoeachvariableislowerthaninInt.Ifthepopulationsize,,islargerthanacriticalvalue,theauthorsshowthattheexpectednumberofgenerationstoconvergenceindistribution,denotedbyNg,anddenedbypNgx=1,isNgOp nforOnemax .12 NgOnforInt: .13 ThelargernumberofgenerationsseenonIntisduetothedierentvariableweightsintheobjectivefunction.Foratruncationrate=0:5,theselectionintherstgenerationisexclusivelybasedonxn,inthesecondgeneration,itisbasedonxn)]TJ/F21 7.97 Tf 6.587 0 Td[(1,etc.Thediscoveryoftheoptimumissequentialinvariablevalues,whereassomelevelofparallelismcanbeachievedonlesshierarchicalobjectivefunctions.

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52 TheexpectednumberofobjectivefunctionevaluationstoconvergenceisNf=Ng:.14Noanalyticalexpressionformwasgivenin Muhlenbeinetal. 1999 .Anapproxi-mationto,bisnowproposedbasedontheinitialrandompopulationsampling,andneglectingvariablevalueslostduringselection.Theprobabilitythatagivenvariablevalueisnotrepresentedinthepopulationisc)]TJ/F15 11.955 Tf 12.402 0 Td[(1=c.Theprobabilitythatthevaluesmakinguptheoptimum,x,haveatleastasampleintheinitialpopulationisPpop=1)]TJ/F28 11.955 Tf 11.955 16.857 Td[(c)]TJ/F15 11.955 Tf 11.955 0 Td[(1 cn:.15ForagivenPpoptypicallycloseto1,thecriticalpopulationsizeisestimatedfromEquation 3.15 ,b=lnn=)]TJ/F22 11.955 Tf 11.955 0 Td[(Ppop lnc=c)]TJ/F15 11.955 Tf 11.956 0 Td[(1Olnn:.16FromEquations 3.12 to 3.16 ,theorderofmagnitudeofthenumberofevaluationstoconvergenceisNfOp nlnnforOnemaxand .17 NfOnlnnforInt: .18 ExperimentalresultsThealgorithmswereappliedtotheproblemsforvedierentnumbersofvari-ablesn=12;20,50,100,and200.TheselectionratiooftheUMDAwaskeptconstantat=0:3asrecommendedin MuhlenbeinandMahnig 1999 .Several

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53 populationsizes0,100,500and1000weretriedinordertoobtainanecientschemeandallowafaircomparisonwithSHC.Twocriteriawereusedtocomparetheperformanceofthealgorithms:thenum-berofanalysesrequiredtoreach80%reliabilitydenedastheprobabilityofnd-ingtheoptimum,estimatedover50independentruns,andthenumberofanalysesneededuntiltheaveragebesttnessreaches98%oftheoptimaltness.Figure 3{10 presentsthenumberofevaluationsto98%ofthemaximumtnessagainstthenumberofvariablesforSHCandfourdierentpopulationsizesofUMDA.Clearly,SHCconvergesfasterthanUMDAforallthenumbersofvariablesinvesti-gated.TheevolutionofthecostofSHCisclosetolinear,whichconrmstheresultsobtainedintheprevioussectionandreportedin Pelikanetal. 2000 fortheOnemaxproblem.ForUMDAwithagivenpopulationsize,thenumberN98ofevaluationsneededtoreach98%oftheoptimaltnessincreasessub-linearly:ttingamodelN98=nlnntotheexperimentaldatayieldedN98=361n0:37lnnfor=500R2a=0:99andN98=650n0:38lnnfor=1000R2a=0:99,conrmingthevalid-ityoftheestimatep nlnnas.Largerpopulationsaremoreexpensive,butsmallerpopulationmayfailtoconvergeforlargen.Thisisthecasewhenapopulationof50individualsisusedtosolvetheMaxA11problemandn50:theaveragemaximumtnessneverreaches98%ofthemaximumtness.Thiscanbeexplainedbythefactthatwhensmallerpopulationsareused,thechanceoflosingparticularvaluesofthevariablesishigher,whichpreventsthealgorithmfromndingtheoptimum,aswasdiscussedinSection 3.2.2 .Theeectofthelossofvariablevaluesforsmallpopulationsisvisibleinthe

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54 Figure3{10:Numberofanalysesuntiltheaveragemaximumtnessreaches98%oftheoptimaltness,maxA11problem.UMDArequirespopulationsofincreasingsizestoavoidprematureconvergenceduetolargetnessvariance.SHCdoesnotexhibitsuchdiscontinuitiesintheperformance. reliability:foreachproblemsizen,thereexistsaminimumpopulationsizebelowwhich80%reliabilityisneverreachedbecauseofprematureconvergenceofthedis-tributions.Thisminimumpopulationsizewasm=100forn=12,m=500forn=20,50and100,m=1000forn=200.Thealgorithmswerethenappliedtothevibrationproblemforthesameveproblemsizes.Thenumberofevaluationsnecessaryfortheaveragemaximumtnessfunctiontoreach98%oftheoptimaltnessisshowninFigure 3{11 .TheresultsaresimilartothoseobtainedfortheMaxA11problem.ThecostofSHCisstillclosetolinearinthenumberofvariables.However,thenumberofevaluationsneededbythetwoalgorithmstoreach98%ofthemaximumtnessissmallerthanontheMaxA11problem.Forinstance,forn=12,SHCneeds64evaluationsonthevibration

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55 problem,against160ontheMaxA11problem.Inthecasen=200,itrequires1,402analysesforthevibrationproblem,against2,923fortheMaxA11problem.Similarly,thenumberofanalysesneededbyUMDAwithapopulationof1000individualsdecreasesfrom4,175analysesto2028analysesforn=12andfrom26,322analysesto17,531analysesforn=200.Thefasterconvergencetowardhightnessregionsforthesecondproblemcanbeexplainedbythefactthatalargepartoftheresponseisgovernedbyoutermostplies,sothatmostofthetnessimprovementcanbeachievedbydeterminingthevalueoftheseinuentialplies.Inaddition,thefactthattheoptimumangleisclosetothecenterofthedomainhelpsSHCbyreducingtheaveragenumberofstepsithastotake. Figure3{11:Numberofanalysesuntiltheaveragemaximumtnessreaches98%oftheoptimaltness,vibrationproblem.Whentheinuenceofsomevariablesonthetnessfunctionisstrongerthansomeothers',theeectofdimensionalityonUMDA'seectivenessisreinforced. Ifthehierarchicalstructureoftheproblemallowsarapidconvergencetohightnessregions,italsocausesnumericaldicultiesforUMDA.Theconvergenceofthe

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56 probabilitydistributionfortheinnermostandoutermostpliesforthecasen=100,m=500arepresentedinFigure 3{12 .Onthishierarchicalproblem,itappearsveryclearlythatthealgorithmproceedsfromtheoutsidetotheinside,startingwiththemoreinuentialvariables,anddeterminingtheinnervariablesonlyattheveryend.Thismechanismisresponsibleforthelossofvariablevaluesfortheless aInnermostply bOutermostply Figure3{12:Evolutionoftheprobabilitydistributions. inuentialinnerplies:intheearlystagesofthesearch,theselectionofgoodpointsisdominatedbytheoutermostvariables:themeanofthetnessofdominatedvariablesisonlyslightlygreaterfortheoptimumvaluethanfornon-optimumvalues,anditsvarianceislarge,aswasremarkedinSection 2.4 .Asaresult,pointswhichcontaintheoptimumvalueoftheoutermostpliesbutnotoftheinnermostpliesgetselected,potentiallyleadingtothedisappearanceofthesevaluesinthedistributioniftoosmallapopulationisused.Weobservedthatlargerpopulationshavetobeusedinordertopreventthelossofvalues.Theminimumpopulationsizesforthisproblemwerem=500forn=12andn=20,m=1000forn=50.Inthecasesn=100andn=200,thealgorithmdidnotreach80%reliabilityforthepopulationsizestested

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57 withinthemaximumnumberof40,000analysesusedinthiswork.SHC,however,isunaectedbytheproblem'shierarchicalstructurebecauseitmerelycomparestwoneighborpoints.Figure 3{13 showsthetwoperformancemea-suresonthetwoproblems.Boththenumberofevaluationsto98%ofthemaximumtnessandthenumberofevaluationsuntil80%reliabilityisachievedarelowerforthevibrationproblemthanfortheMaxA11problem:thelocationoftheoptimuminacenterareaofthedesignspacemakesthevibrationproblemeasierforSHC. Figure3{13:ComparisonoftheperformanceofSHContheMaxA11problemandthevibrationproblem. Thisstudyrevealsthatthepotentialbenetsofaprobabilisticoptimizationalgorithmdonotlieintheirperformanceinthefaceofincreasingdimensionality:ifitistruethatforagivenpopulationsize,UMDAwouldasymptoticallyoutperformSHC,whichneedsclosetolinearlyincreasingnumbersofrandomtrialstondtheoptimum.Tomaintainasucientaccuracyofthestatisticalmodel,largepopulationshavetobeusedinUMDA,therebyneutralizingtheadvantageofthatmodel.Like

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58 otherEAs,EDAsarecostlyalgorithms.Theiradvantageisobservedonproblemsthatcannotbehandledbysimpleralgorithms,suchasmultimodalproblems,aswillbedemonstratedinSection 3.3 .3.3InvestigationofpossiblemodicationstoUMDA:memoryandmutationWehaveseeninSection 3.2.2 thatUMDAvariantsthathadasmallpopulationsizeandalargeselectionpressureexhibitedthefastestconvergence,butthattheyalsodisplayedapoorreliabilitybecausewhenavaluedisappearedfromthedistributionatagiveniteration,norecoverymechanismallowedittobereintroducedatalatergeneration.Whenthelackofdiversityinjectionmechanismisassociatedwithaveryselectiveschemethatconvergesrapidlytoasmallnumberofvalues,thealgorithmbecomesstronglyexploitative,andprematureconvergenceisverylikely.Thereliabilityofthesevariantsmaybeimprovedbythreedierentstrategies:slowingdowntheconvergenceoftheprobabilitiesusingmemory,providinganallelerecoverymechanismintheformofmutation,orspecicallynotallowingprobabilitiestovanish.Thissectioninvestigatestheeectofmemory,mutation,andofimposinglowerboundsontheprobabilitiesontheperformance.Thetestswereperformedonthe10-variableA11maximizationproblemintroducedinSection 3.2.1 .3.3.1MemoryAsimplewayofpreventingprematurelossofdiversityduetosamplingerrorsistouseaconservativeupdatingmethodthatmakesitpossibletoadjusttheinuenceofincomingobservations,asproposedby Baluja 1994 .Thenewprobabilitydistri-butionpt+1xkisobtainedasalinearcombinationoftheolddistributionptxkand

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59 theobservedfrequenciesfk:pt+1ki=1 m+1)]TJ/F22 11.955 Tf 5.48 -9.684 Td[(mptki+fki;.19wheremisamemory"parameter:whenm=0,thenewprobabilitiesaresimplythemarginalfrequenciesstandardUMDA,whenm6=0,thealgorithmmitigatespotentialsamplingerrorsinthefrequenciesbygivingthemasmallerweight.Thevalueofmreectstheuser'scondenceinthefrequencies'accuracy.Toinvestigatetheeectofmemory,weconsideredtwovariantsofUMDA:A1,thebestschemefoundinSection 3.2.2 f=50,=166gandA2,thealgorithmthatdisplayedthefastestinitialconvergencef=10,=33gseeFigure 3{5 .Sixvaluesofmweretested:m=0,1,2,5,10,and20.ThereliabilityofthetwovariantsforthevevaluesofmemoryarecomparedtothebasicUMDAinFigure 3{14 .Not Figure3{14:EectofmemoryonUMDAwithtwoparametersettings,maxA11problemn=10:f=10,=33gsolidlinesandf=50,=166gdashedlines.

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60 surprisingly,addingmemorytothebestschemeA1causedthereliabilitytodeterio-rate:thepopulationsizeshadbeenadjustedforthesimpleUMDA,buttheyarenolongerappropriatewhensubstantialmodicationsaremadetothealgorithm,hencethedecreaseinreliability.Incontrast,A2'sreliabilityimprovesdramaticallywhenmemoryisused:thenalreliabilityincreasesmonotonicallywithm.However,thisimprovementisachievedatthecostofconvergencevelocity,aslargememoryclearlyaddsinertiatothesearchbygivingalargeweighttopastobservations.Asaresult,thebenetofmemoryisdebatable,andincreasingpopulationsizesmaybeamoreeectivewayofpreventingdistributiondegeneracy,assuggestedbythecomparisonbetweenthememorylessA1andtheA2versions,whichincorporatememory.3.3.2MutationAcustomarystrategyforpreventingprematureconvergenceinevolutionaryalgo-rithmsistouseaperturbationoperatorcalledmutation.Inthecontextofestimationofdistributionalgorithms,twoapproachescanbeadoptedtoimplementmutation:onecaneitherapplytheperturbationstothepointsordirectlytothedistribution.Fewinstancesofmutationareprovidedintheliterature. Baluja 1994 proposedamutationconsistinginshiftingtheprobabilitiesbyaxedvaluewithagivenprob-ability.Heconcludedthatwhilesomeimprovementcouldbeachieved,theoperatorwasnotascriticalasinstandardgeneticalgorithms.Inthiswork,wechosethealternateapproachofapplyingperturbationstothepointsgeneratedbytheunper-turbedprobabilitydistribution.Indeed,inEDAs,thestatisticalmodelreectsourknowledgeofpromisingregions,hencethereisnoreasontodegradethisinformation

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61 arbitrarily.Weprefertoprovideaseparatediversityinjectionmechanism:iftheproposedperturbationisvalidatedthroughselection,itwillbeincorporatedintothemodelatthenextiteration.Themutationoperatorusedinthisworkwasanadjacentmutation",wherethevalueofeachvariablexkwaschangedwithprobabilitypmtooneofthetwoneighboringvalues 6 forinstance,30couldbechangedinto15or45withequalprobabilitiesifthevalueswerechosenfromf0;15;30;45;60;75;90g.UMDAwasappliedtothe10-variableA11maximizationproblempresentedintheprevioussectionswithsevenvaluesofthemutationrate:pm=0,0.005,0.01,0.02,0.05,0.1,0.2,and0.3.Asintheprevioussection,twoschemeswereconsidered,A1,thebestschemefoundinSection 3.2.2 f=10,=33gandA2,thealgorithmthatdisplayedthefastestinitialconvergencef=50,=166g.Figure 3{15 showstheinuenceofthemutationrateonthereliabilityofthealgorithmforthetwovariants.Likememory,mutationhadanegativeimpactonA1,becausethevaluesofandchosentomaximizethereliabilityofUMDAwerenolongerthebestinamodiedcontext.Incontrast,usingmutationincombinationwithA2ledtoadramaticimprovementofthereliabilityofthealgorithm:whiletheoriginalalgorithmhadaverylowmaximumreliabilityof16%,evenwithalowmutationrateof0.005the 6Theboundaryvalues0and90requiredaspecialtreatment:whenmutationwasappliedtothesevalues,thevariablehadequalchancesofremainingunchanged,orbeingshiftedtothenextvalue5foraninitialvalueof0,75foraninitialvalueof90.

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62 Figure3{15:InuenceofmutationonthereliabilityfortwoUMDAvariants:thebestschemewithoutmutationf=50,=166g,andtheonethatdisplaysthefastestinitialconvergencef=10,=33g.Mutationdramaticallyimprovestheformervariant'sperformancebypreventingcompletedisappearanceoftheoptimalvariablevalues,butaectsthelatter'sreliabilitynegatively,becauseitperturbstheestimationofdistributions. algorithm'sreliabilityreaches80%inapproximatelythesamenumberofevaluationsasA1.Thereliabilitycanbefurtherimprovedbyincreasingpm,thebestperformancebeingobservedforpm=0:2.Beyondthisvalue,asharpdeclineinthereliabilitytakesplace.Thelowsensitivityofthereliabilitytothemutationratepmoverawiderangeofvaluesconstitutesapositiveattribute,asitreducestheamountoftuningnecessarytousethealgorithm.Unlikememory,mutationdidnotaecttheconvergencevelocitysubstantially.whenpmislow,mutationactsasabackgroundoperatorthatcompensatesfordetri-mentaleectsofworkingwithanitepopulation.Theroleofmutationintheopti-mizationcanbeobservedinFigure 3{16 :intherstveiterations,itpreventstheoptimalvaluex1=0fromdisappearingfromthedistribution;thealgorithmthen

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63 identiesitastheoptimalvaluethroughtheprocessofselectionanddistributionestimation. Figure3{16:Eectofmutationforpm=0:005.Thankstomutation,theoptimalvaluex11=0,whichwouldotherwisehavedisappearedfromthedistribution,isreintroducedthroughmutation.Sincetheperturbationlevelislow,thedistributioneventuallyconvergestotheoptimalvalue. 3.3.3BoundontheprobabilitiesWehaveseeninprevioussectionsthatoneofthemajorconcernsinestimationofdistributionalgorithmswastomaintainthealgorithm'sabilitytovisitanypointinthedesignspace.Thiscanbemosteasilyachievedbyformallypreventingmarginalprobabilitiestofallbelowathreshold.Letpki=pxk=ai.Thefollowingprocedurewasimplementedforeachmarginaldistributionpxk: 1. ifaprobabilitypkiissmallerthan,itissetto; 2. thesumofthedierences)]TJ/F22 11.955 Tf 12.878 0 Td[(pkiofallthecorrectedprobabilitiesisevenlydistributedovertheothervalues.

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64 Theeectofimposingalowerboundonthemarginalprobabilitieswasinvesti-gatedforthetwoUMDAschemesusedinprevioussections.ThereliabilityforthreevaluesofisshowninFigure 3{17 valuesofaregivenasfractionsoftheuniformprobabilitypki=1=c.Asfortheotherdiversityinjectionmechanismspresented Figure3{17:Eectofimposingalowerboundontheprobabilitiesonthereliability. previously,imposingboundsonthemarginalprobabilityhadanadverseeectonthereliabilityofthealreadyclosetooptimalschemeA1.However,ithadadramaticimpactonA2:withvaluesofbetween1 40cand1 10c,thereliabilityreached100%inlessthan2,000evaluationswithoutappreciabledeteriorationoftheconvergencevelocity.3.3.4ElitismAstrategyoftenusedtoguaranteeamonotonicincreaseofthetnessinevo-lutionarycomputationiselitism,whichconsistsinautomaticallycopyingthebest

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65 Figure3{18:Eectofanelitiststrategyonreliability. parentinthechildpopulation.Theunderlyingprincipleisthatthepopulationcon-stitutesthememoryofpastobservations,hencegoodindividualsencounteredduringthesearchshouldnotbelost,astheyareevidenceforgoodregions.IntheEDAframework,thenotionofindividualislessimportant,astheutilityofpopulationsofpointsisonlytocharacterizethedistributionofpromisingregions;afterinformationhasbeenextractedfromgoodpoints,thepopulationbecomesuseless.However,someauthorshaveproposedreplacementstrategiesallowingaportionofapopulationtobekeptforthenextiteratione.g. ChoandZhang 2002 ; BosmanandThierens 2000 .TheinuenceofelitismwasinvestigatedforthetwoUMDAschemesA1andA2:ateachiteration,)]TJ/F15 11.955 Tf 12.143 0 Td[(1werecreatedbysamplingfrompx,andthebestpointofthepreviousgenerationwasaddedtoobtainapopulationofpoints.Thereliabilityofelitistalgorithmsiscomparedtothatofnon-elitistalgorithmsinFigure 3{18 .ElitismdidnothelpforeitherofthetwoUMDAschemesconsidered:inthecaseofA1,the

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66 populationsizeswerelargeenoughtoguaranteeagoodaccuracyofthedistribution,sothatnoinformationaboutgoodregionswaslostandelitismwasnotneeded;inthecaseofA2,copyingthebestpointofapopulationinthenextiteration'spopula-tionreinforcedthealgorithm'stendencytoprematureconvergence,whichcausedthereliabilitytofurtherdeteriorate.3.3.5ConclusionoftheparameterstudyInthissection,fourpossiblemodicationstotheoriginalunivariatemarginaldis-tributionalgorithmwereinvestigated.Themaingoalwastoprovideamechanismtopreventprematureconvergenceofthesearchdistribution.Threedierentapproacheswereproposed:memory,mutation,andalowerboundonthemarginalprobabilities.Allthreestrategiesyieldedsubstantialperformanceimprovement.Inparticular,theymadetheuseofsmallpopulationsviable,therebyallowingsavingsoffunctioneval-uations,eitherbyallowinglostvaluestoberecovered,orbypreventingthelossofvariablevalues.Thethreeapproachessharethecommoncharacteristicthattheycanhampernalconvergencewhenusedtoomassively.Thiscallsforadaptivestrategiestodecreasetheinuenceoftheseperturbingoperatorsovertime,whichisbeyondthescopeofthisstudy.Finally,astudyoftheeectofelitismwascarriedoutforcompletthisstrategycommonlydidnotleadtoaperformanceimprovementforthetwovariantsinvestigated.3.4ComparisononthreeproblemsInthissection,wecomparetheperformanceofUMDAtotwootherstochasticoptimizationalgorithms:astochastichill-climber,andastandardgeneticalgorithm.

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67 Thethreealgorithmsareappliedtothreelaminateoptimizationproblemscarefullychosentoassesstherelativeperformanceofthealgorithmsonrepresentativetnesslandscapes.3.4.1PresentationofthealgorithmsAnUMDAwithtruncationselectionofratio=0:3wascomparedtotwostochasticoptimizationalgorithms:astochastichill-climberSHC,presentedinSec-tion 3.2.3 ,whichisapoint-basedrandomsearchalgorithm,andastandardgeneticalgorithmGAseeAppendix D foranintroductiontogeneticalgorithmswithrankproportionalrouletteselectioni.e.linearrankingselectionandtwo-pointcrossoverwithprobabilitypc=1:0.NoelitiststrategywasimplementedinUMDAandGA:wewanttocomparethefundamentalsearchmechanismsofthealgorithms.ThestudywasprimarilyfocusedonUMDAandSHC:alimitedparameterstudywasperformedtoobtaingoodvaluesofthemutationratepmandpopulationsize.GA'sperfor-manceforthebestUMDAsettingisprovidedtoallowreadersmorefamiliarwiththatalgorithmtoassessUMDA'sperformance.3.4.2ConstrainedmaximizationoftherstnaturalfrequencyTherstproblemwasmaximizingtherstnaturalfrequencyofasimplysup-portedgraphite-epoxylaminatedplateoflengthL=50"andwidthW=15"subjecttoaconstraintontheeectivePoisson'sratioleu,withl=0:48andu=0:52.TherstnaturalfrequencyisgivenbyF=2 p mhr D11 L4+2D12+2D66 L2W2+D22 W4.20

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68 wherehisthetotallaminatethickness,mdesignatesthemassdensity,andtheDij'sarethebendingstinesscoecients.TheeectivePoisson'sratioisgivenbye=A12 A22=U1)]TJ/F22 11.955 Tf 11.956 0 Td[(U2V1+U3V3 U4)]TJ/F22 11.955 Tf 11.955 0 Td[(U3V3.21wherethein-planelaminationparametersV1andV3areobtainedbyV1=1 nnXk=1cos2k,V3=1 nnXk=1cos4k:.22ThematerialpropertiesusedforthisproblemareshowninTable 3{3 Table3{3:Materialpropertiesofgraphite-epoxy. LongitudinalmodulusE12:18107psiTransversemodulusE21:38106psiShearmodulusG121:55105psiPoisson'sratio120.26Weightdensity0.057lb/in3Plythicknesst0.005in Noneofthethreealgorithmsaccommodatesconstraints,thereforeapenaltyap-proachwasused,wherethetnessfunctionFpofinfeasibledesignswasdecreasedinproportiontotheconstraintviolation:Fp=8>><>>:Fxifgx0feasibleFx)]TJ/F22 11.955 Tf 11.955 0 Td[(pgxifgx>0infeasible.23wherepisthepenaltyparameterwhosevaluewasadjustedempiricallytoensurethatthealgorithmyieldedfeasibledesigns.Weusedp=2103forthisstudy.Theconstrainttermgxwasdenedasgx=max1)]TJ/F22 11.955 Tf 13.151 8.088 Td[(ex l;ex u)]TJ/F15 11.955 Tf 11.955 0 Td[(1:.24

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69 TheconstraintonthePoisson'sratioforcesthepointstoremaininanarrowchannelinthedesignspace,whichmakestheproblemparticularlydicultforhill-climbingalgorithmsbecausemanyoftherandomperturbationsresultininfeasibledesigns.Clearly,thisproblemdidnotsatisfythevariableindependenceassumption,asthevalueofonevariableinuencestheoptimalvalueofothervariablesthroughtheconstraint.Twonumbersofvariableswereconsidered:n=8andn=15.Withoutthecon-straint,theoptimalorientationwouldbe90foralltheplies.TheeectivePoisson'sratiowouldthenbee=21=0:0165.ThePoisson'sratioconstraintforces30,4560,and75pliesintotheinnerlayersofthelaminate,wheretheyaretheleastdamagingtothefrequency.Theoptimum 7 forn=8was[902=75=455=30]s,whichhasarstnaturalfrequencyofF=670HzandaneectivePoisson'sratioofe=0:482.Forn=15,theoptimumwas[904=75=602=455=305]s,withF=1;262:6Hzande=0:481.Forn=8,twopopulationsizes,=20and=50,andthreemutationrates,pm=0:1,pm=0:2andpm=0:3pervariableweretriedforUMDA.Inordertoselectthebestscheme,thereliabilityreachedat2000evaluationswasusedascriterion.Thehighestreliability88%wasachievedwith=20andpm=0:2.AsimilarparameterstudywasconductedforSHCwithpmrangingfrom0.1to0.5,andthehighest 7Forthisproblem,asforalltheproblemsaddressedinthiswork,thebestsolutionfoundinalltherunsperformedisconsideredastheoptimum,unlesstheglobaloptimumisknownapriori.

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70 reliabilitywasachievedforpm=0:4.Figure 3{19 comparesthemeanbesttnessandreliabilityofthethreealgorithmsforthesebestsettingsthesamepopulationsizeandmutationratewereusedforGAandUMDA,toensurecomparablevariabilityofthesearchdistributions.SHCconvergesfasterthanGAandUMDAtohigh a bFigure3{19:MeanbesttnessaandreliabilitybforSHC,UMDA,andGAfortheconstrainedmaximizationoftherstnaturalfrequencyofalaminatedplaten=8.Whenthenumberofvariablesissmall,SHCcanprogressalongthetunnel,anditsperformanceiscomparabletothatofUMDA. tnessregions,asshowninFigurea,howeverthisadvantagedoesnottranslateintoahigherreliabilitythanUMDA.SHC'sfailuretonalizethesearchcanbeattributedtothehighmutationrate,whichisanindicationthatanadaptiveschememayimprovetheperformance.GA'sapparentpoorreliabilityistheresultofaweakerselectionprocedurelinearranking,cf.Section 2.4 .However,consideringthatitisonethemostcommonlyused,theobservedbehaviorcanarguablyberegardedasrepresentativeofatypicalGA.Clearly,bothSHCandUMDAndtheoptimummorereliablythanGAonthisexample.Forn=15,apopulationsizeof=100individualsandamoderatemutation

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71 probabilitypm=0:1werechosen.Thelargerpopulationisjustiedbecausemoreindividualsareneededtoensurethatallvariablevaluesarepresentinthepopulation.ForSHC,thebestmutationprobabilitywasfoundtobepm=0:2testedvalueswerepm=0:1;0:2;0:3;0:4;0:5.Figure 3{20 comparestheperformancecriteriaforthethreealgorithms.Thistime,UMDAseemedtobenetfromtheuseofaglobal a bFigure3{20:MeanbesttnessaandreliabilitybforSHC,UMDA,andGAfortheconstrainedmaximizationoftherstnaturalfrequencyofalaminatedplaten=15.Whenthenumberofvariablesissucientlylarge,progressingalongthechannel"becomesincreasinglydicultforSHC,andUMDA'sglobalapproachbecomesmoreeective. probabilisticmodel,whichallowsittoescapelocalminimaandwasabletoreliablyndtheglobaloptimum:after10,000evaluations,thereliabilityoftheoptimizationhadreached64%,whereastheSHConlyfoundtheoptimumin30%oftheruns.In70%oftheruns,SHCconvergedtoahighqualitysolutionbutfailedtoyieldthetrueoptimum.Forinstance,oneofthesolutionswas[906=60=4510=30]s,whichhadatnessofFp=1;279:3F=1;257:9,=0:481.Inordertoobtaintheglobaloptimum[904=75=502=455=305]sFp=1;262:6,F=1;262:6,=0:481,sixvariableshavetobemutated.However,allsinglemutationsleadtoareduction

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72 inthetnessfunction,eitherbecausetheymakethedesigninfeasiblevariable5orbecausetheydecreasethevibrationfrequency.Consequently,multiplemutationsmustoccursimultaneouslyforthetnessfunctiontoimprove.Theprobabilityofthateventdecreasesasnincreases,thusmakingfurtherprogressofSHCunlikely.Theseresultsagreewith HeandXin 2002 inwhichanSHCwasshowntohaveanexponentialtimecomplexityforamultimodalfunction,whilethecostofapopulationbasedEAwasonlypolynomial.3.4.3MinimizeA66SHCismisledbylocalminima,whiletherelationshipbetweenUMDAandtheobjectivefunctionismorecomplex.Itisneverthelessknownthat,iftheobjectivefunctionisseparable 8 andthepopulationsizeislargerthanacriticalvalue,UMDAconvergestotheglobaloptima MuhlenbeinandMahnig 1999 .AnexampleofsuchanobjectivefunctionwherethereliabilityofUMDAtendsto1whilethatofSHCisnearly0istheminimizationofthein-planeshearstinessofacompositelaminate,A66,overplyorientationsthatareboundedbetween0and75,min0xi75A66;.25whereA66=U5h)]TJ/F22 11.955 Tf 11.955 0 Td[(U3nXi=1ticos4xi:.26 8Amoregeneralresultisgivenin MuhlenbeinandMahnig 1999 wherethecon-vergenceofFactorizedDecompositionAlgorithm"FDAtotheoptimaisprovedforadditivelydecomposedfunctions.SeparablefunctionsareaspecialcaseofadditivelydecomposedfunctionsandUMDAisthecorrespondingsimpliedFDA.

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73 Theglobaloptimumisxi=0,i=1;n.Replacinganyofthexiwith75createsalocaloptimumwhosebasinofattractionstartsatx=45.Forann-dimensionalcase,thereare2nlocaloptima.Numericalexperimentswithn=12averagedover50runsconrmthattheSHCreliabilityis0itistheoretically=7512=2:10)]TJ/F21 7.97 Tf 6.586 0 Td[(3whilethatofanUMDAwithm=500reaches1after3,500analyses.Figure 3{21 showstheevolutionoftheprobabilitydistributionoftheoutermostplypx1.Itisinterestingtonotethatintheearlystagesofthesearch,boththeprobabilityof75andtheprobabilityof0thetwolocaloptimaincrease.Butafterabout2,000analyses,theprobabilityof75startstodecreaseandthealgorithmconvergestotheglobaloptimum. Figure3{21:Evolutionofthemarginaldistributionpx1,UMDA,maxA66problem.

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74 3.4.4StrengthmaximizationManypracticallaminateoptimizationproblemsexhibitamultimodalrelation-shipbetweenthevariablesandthetnessfunction.Atypicalexampleofsuchbehav-ioristhemaximizationofthestrengthofalaminate.Inthissection,weconsideredtheproblemofmaximizingtheloadfactors,usingtherst-ply-failurecriterionbasedonthemaximumstrain Gurdaletal. 1998 ,Chapter6,foraglass-epoxylaminatesubjectedtothein-planeloadingNx=)]TJ/F15 11.955 Tf 9.299 0 Td[(1000103N/m,Ny=200103N/m,Nxy=400103N/m:maximizes=nmink=1minult1 1k;ult2 2k;ult12 12kwheretheloadfactorsisthecoecientbywhichtheloadhastobemultipliedforthestructuretofail.ThematerialpropertiesusedforthisproblemareshowninTable 3{4 .Thetotalthicknessofthelaminatewash=0:02m. Table3{4:Materialpropertiesofglass-epoxy. LongitudinalmodulusE169:0GPaTransversemodulusE210:0GPaShearmodulusG124:5GPaPoisson'sratio120.31MaterialinvariantU131.58GPaMaterialinvariantU227.91GPaMaterialinvariantU36.48GPaMaterialinvariantU49.63GPaMaterialinvariantU510.98GPaTensilestrengthXt500.0MPaCompressivestrengthXc410.0MPaTensilestrengthYt35.0MPaCompressivestrengthYc110.0MPaShearstrengthS70MPa Weconsideredthecasen=8designvariables.Thelaminateissubjectedtoan

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75 in-planeloading:Nx=)]TJ/F15 11.955 Tf 9.298 0 Td[(1000kN/m,Ny=200kN/m,Nxy=100kN/m.Forthisproblem,theoptimumlaminatewas[06=9010]soritspermutations,andtheoptimumloadfactorwass=5:39.Dependingontheorientationofthebers,oneofthethreepossiblefailuremodesberfailure,matrixcracking,shearfailurebecomescritical.Thecombinationofthesethreefailuremodesresultsinamultimodaltnessfunction 9 a bFigure3{22:ReliabilityaandmeanbesttnessbofUMDAandSHCforthestrengthproblem.UnlikeSHC,whichwastrappedinlocaloptima,UMDAwasabletondtheglobaloptimumreliably.WhenthemaximaoftheobjectivefunctionarefarapartintheHammingspace,SHCisnotabletoescapelocalmaxima,henceitspoorreliability.Incontrast,UMDA'sglobalsearchstrategyallowsittolocatehighfunctionevaluationregionsandtoreliablyndtheglobalmaximum. ForUMDAthepopulationsizewas=50andamoderatemutationrateofpm=0:1pervariablewasimplementedtopreventprematureconvergence.SeveralvariantsofSHCwerecomparedtoobtainthebestcompetitortoUMDA.Forthis 9Notethatitsisnotnecessarytoconsiderallthefailuremodesfortheresponsetobemultimodal,asindividualfailuremodescanbemultimodalforspeciccombi-nationsofmaterialpropertiesandloading.

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76 problem,thesameadjacentmutationoperatorwasimplementedandthebestratewaspm=0:2thesamevaluesofandpmwereusedforGA.BoththereliabilityFigure 3{22 -aandthemeanbesttnessFigure 3{22 -bclearlyshowthesuperiorityofUMDAforthisproblem.ThereliabilityofSHCincreasesfasterthanthatofUMDA,butculminatesat36%,whileUMDAwasabletoconvergereliablytotheoptimum.SHCoftenconvergedtolocaloptima[08=908]s,s=4:97,[06=30=908]s,s=4:47or[08=30=906]s,s=4:02andwasnotabletoescapeevenwhenlargemutationrateswereusedbecauseunlikelycoordinatedmutationswereneededtoreachthebasinofattractionoftheglobaloptimum.3.5ConclusionUnivariatemarginaldistributionalgorithmsconstitutethesimplestformofesti-mationofdistributionalgorithms.Theyestimatedistributionsbasedontheassump-tionthatthevariablesarestatisticallyindependentinthepopulationsofselectedpointsbuttheyareoftenappliedtoproblemsthatviolatethisassumption.Inthischapter,theinuenceofmajorcomponentsofthealgorithmwasinvestigatedonalaminateoptimizationproblem.Theselectionpressuredeterminestheconvergencevelocity,whilethepopulationsizeaectsthevarianceofthemarginalfrequencies,hencetheaccuracyofthesearchdistribution.UMDA'sasymptoticbehaviorwascom-paredtothatofahill-climbingalgorithm.ThestudyconcludedthattheadvantageofUMDAdoesnotlieinpureconvergencevelocity.Instead,itsglobalsearchcapa-bilitiesbecomebenecialforfunctionsthatexhibitnarrowchannels,ormultimodallandscapes,asdemonstratedonthreelaminateoptimizationexampleproblems.

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CHAPTER4THEDOUBLE-DISTRIBUTIONOPTIMIZATIONALGORITHMInthischapter,weproposeaphysics-basedmethodforincorporatingvariabledependenciesintotheprobabilisticmodelofpromisingregions.Theapparentcom-plexityofthedistributionofselectedpointscanoftenbeexplainedasthejointactionofasmallnumberoflatentvariables.Thedistributionofselectedpointscanbere-constructedbythecooperationofsimplemodelsoftheprimaryvariablesandofthelatentvariables,usedasauxiliaryvariables.ThisrepresentationofthedistributionofpromisingpointsisthebasisoftheDouble-DistributionOptimizationAlgorithm,introducedhere.4.1MotivationInChapter 3 ,thedistributionofselectedpointswasapproximatedbyaproductofunivariatemarginaldistributions,whichneglectedanystatisticalvariabledepen-dencies.Whilethisrepresentationyieldedcomparabletosuperiorperformancethanastandardevolutionaryalgorithm,itmaynotbeappropriateforproblemswithstrongvariableinteractions.Figure 4{1 illustratestheimpactofthechoiceofaunivari-atemodelwhenthedistributiongeneratedbyselectionismorecomplex.Inthe2DproblemconsideredwhichistheproblemofmaximizingtherstnaturalfrequencysubjecttoaconstraintonPoisson'sratiointroducedinChapter 3 ,thecontoursoftheobjectivefunctionassumeroughlytheshapeofanarrowridge.Afteruniformsamplinginthe1;2-plane,andapplicationoftruncationselection,thepromising 77

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78 a bFigure4{1:Selectedpointsaandunivariatedistributionbfor1;22f0;5;10;:::;85;90g2fortheconstrainedvibrationproblemofChapter 3 .Whenthedistributionofselectedpointsisapproximatedbyaproductoftheunivariatemarginaldistributionsp1andp2,high-probabilityregionsdonotcoincidewithhigh-tnessregions. pointsaredistributedasshowninFigure 4{1 -a.Ideally,onewouldwanttosamplefromthehighevaluationregionsmarkedbytheselectedpoints.However,whenthejointdistributionp1;2isapproximatedbyaproductoftheunivariatemarginaldistributionsp1andp2,newpointswillbecreatedaccordingtothedistribu-tionwhosehistogramisshowninFigure 4{1 -b.Clearly,thehighprobabilityareasdenedbythatdistributiondonotcoincidewithhighevaluationregions, 1 sothatsomeknowledgeaboutoptimalregionscontainedintheselectedpointswillbelost 1Inaunivariatemodel,maximumprobabilityareasareparalleltotheaxes.Forexample,forx=x1;x2,themaximumprobabilityatagivenvalueofx2ismaxx1px1px2=px2maxx1px1=px2px1,wherepx1isthemaximumofthemarginaldistributionpx1:thevalueofx1thatmaximizespx1;x2doesnotdependonx2.

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79 forthenextEDAgeneration.Onewayofimprovingthedistributionaccuracywouldbetouseahigher-orderstatisticalmodelthatincorporatestheconditionalprobabilitiesp2j1.Inhigherdimensions,thisapproachcouldbegeneralizedbyusingaBayesiannetworktorep-resentallconditionalprobabilitiespiji,whereidesignatestheparentsofthevariablei,asproposedby Pelikanetal. 1999 intheBayesianOptimizationAlgo-rithm.However,thisapproachpresentsthreestronglimitations: thenumberofparametersrequiredtodescribehigh-orderdependenciesinthedistributionincreasesexponentiallywiththenumberofvariablesncn)]TJ/F15 11.955 Tf 13.229 0 Td[(1parametersarenecessarytoexpressthefulljointdistributionwheneachofthenvariablescantakecvalues.Thecomputationalcostassociatedwiththeestimationofthoseparametersmaybecomeprohibitiveasnbecomeslarge; moreimportantly,thestandarderrorassociatedwiththeparametersincreaseswiththeexibilityofthemodel.Whileaexiblemodelcanasymptoticallyapproximatecomplexdistributionsafortiorisimpledistributions,obtaininganaccurateestimationoftheparametersmayrequireaverylargesamplecf.Section 2.5.3 .InthecontextofEDAs,thisimpliesthatlargepopulationmayhavetobeused,sothatlargerprogressmaybeaccomplishedateachiteration,butatthecostofmanyfunctionevaluations; lastly,usingatoohigh-ordermodelviolatesthepremisesofoptimization,whichisbasedontheassumptionthatthereexistsanunderlyingstructurepresentinthedistributionpx:thegoalistoinferandusethisstructuretondtheoptima.Thisimpliesatrade-obetweenaccuracyandgeneralizationfromthe

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80 sample:high-ordermodelswillprovideanaccurateestimationonthetrainingset",butextrapolationtootherregionsofthedesignspacewillbepoor.Alternativeapproachesareusedroutinelybystatisticiansintheeldofex-ploratorydataanalysis.Theideaistosimplifythedatabylookingforstructureinthedistributions,aswillbepresentedinthenextsection.4.2Principles4.2.1IdenticationofdependenciesanddatasimplicationAtypicalexampleofaneorttodistinguishbetweendatastructureandran-domvariationsisexploratoryfactoranalysisEFA,wherethegoalistoidentifyunderlyinglatentfactorsthatdeterminethedistributionofapopulation,andusethemtodescribethedistribution,therebyasshowninFigure 4{2 .Bysupposingthat Figure4{2:Interpretationofvariabledependenciesasthejointactionofhiddenvari-ablesV1;:::;Vm.Evenifthehiddenvariableshaveasimpleprobabilitydistribution,thedistributionoftheobservedvariablestheX'scanbecomplex. thepossiblycomplexobserveddistributionistheresultofjointactionsofasmallnumbermoflatentvariables,whosedistributionissimple,oneisabletosimulta-neouslyreducethenumberofmeaningfulvariables,aswellasthecomplexityofthedistributionthelowdimensionaldistributionofindependentlatentvariablesareeas-iertoestimatethanthejointdistributionoftheobservedvariables.Suchmethods

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81 wereimplementedby Shinetal. 2001 ,usingseverallatentvariablemodels:theymodeledthedistributionofselectedpointsbyHelmolzmachines,andprobabilisticprincipalcomponentanalysisPPCAtorepresentapparentlycomplexdistributionsastheresultofthejointactionofseverallatentvariables,whosedistributionissimplemultivariatenormal.Thisgeneralmethodisappropriatewhennothingisknownaboutthestructureofthedistribution,howeveritonlyallowssimpletypicallylinearrelationshipstobeidentied,andlearningeventheserelationshipshasacomputationalcost.Inmanysituations,aprioriknowledgeabouttheproblemisavailable.Thebenetsofprovidingthedistributionstructurewasinvestigatedby Baluja 2002 ,whoshowedthatitledtodrasticperformanceimprovement.Inthiswork,weproposeafullydierentapproachforincorporatinginformationaboutthestructuretoenhancetheaccuracyoftheestimateddistributionbydirectlyprovidingthealgorithmwithitsstructure.Inmanyinstances,variabledependenciesamongselectedpointsoftenreectthefactthattheoverallresponseofthesystemisreallyafunctionofintegralquantities,sothatmanycombinationsofthedesignvariablescanproducethesameresponse.Forexample: thedimensionsofthesectionofabeamdetermineitsexuralbehaviorthroughthemomentofinertiaI, theaerodynamicpropertiesofavehiclearecapturedbythedragcoecientCD, theowthroughaporousmediumisdescribedbythepermeabilityk, etc.

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82 Thesequantitiesareofteninexpensivetocalculate,andtheirnumberislimitedandinsensitivetothenumberofdesignvariables.Theproposedstrategycombinesanindependentmodelfortheprimaryvariablesxk,andacomplementarymodeloftheauxiliaryvariablesfVasshowninFigure 4{3 .Thedistributionofauxiliaryvariablesintroducesinformationaboutstatisticalvariabledependenciesintothees-timatedmodelofprimaryvariables 2 Figure4{3:Incorporationofvariabledependenciesthroughauxiliaryvariables. 4.2.2SamplingfromtwodistributionsNowthattheprincipleshavebeenestablished,theactualprocedureimplementedintheoptimizationalgorithmneedstobespecied.ThegoalistodeviseastrategybywhichpopulationsofpointsreectingthetwodistributionspxandpVcanbe 2Arelatedstrategyconsistinginusingasimplemodel,whichisincrementallyimprovedbyincorporatinginformationaboutthedistributionofparticularcombi-nationsofthevariablesisusedinstatistics,namelyinestimationofdistributionbyprojectionpursuit Friedmanetal. 1984

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83 created 3 .Theparticularstrategyadopteddeterminestherelativeinuenceofeitherdistributioninthecreationprocess.Threedierentschemeshavebeeninvestigatedinthiswork:inallcases,thegeneralprincipleistocreatealargepoolofcandidatepointsbysamplingfrompx,and2retainpointsbasedonthedistributionpV.Thethreemethodsdierintheproceduretheyusetodecidewhichpointstokeep.DeterministicacceptanceTheprobabilitypVofeachofthecandidatepointsintheauxiliaryvari-ablespaceiscalculated.Then,thepointshavingthehigherevaluationsofpVareacceptedasthenewpopulation.Preliminarytestsindicatedthatthisstrategyproducedaveryconservativealgorithm,whichdidnotallowtheexplorationofnewregions.ProbabilisticacceptanceTheprobabilitypVofeachofthecandidatepointsintheauxiliaryvariablespaceiscalculated.Then,pointsaredrawnamongthepoolofcandidates,withaprobabilityproportionaltopV:pVi=pVi Pj=1pVi: 3Unlikestandardfactoranalysismethods,whicharegenerativebecausetheob-servedvariablesareexplicitfunctionsofthehiddenvariables,theproposedtwo-distributionapproachdoesnotprovideadirectwayofgeneratingsamples.

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84 Thisapproachisattractive,asthenalprobabilitydistributionunderthispro-cedureissimplyobtainedastheproductofthetwodistributions:pnalx1;:::;xn/nYk=1pxk!pV1x1;:::;xn;:::;Vmx1;:::;xn:.1Thismethodissimpletoanalyze,andmakesaclearparallelwithwell-establishedmethods,suchasprojectionpursuitdensityestimation Friedmanetal. 1984 .How-ever,theapproachpresentstwodisadvantages:rst,eventhoughtheresultingdistri-butionconstitutesacompromisebetweenthetwodistributions,therespectivecon-tributionofeachdistributiontothenaldistributionisarbitrarilyxed.Thereisnotheoreticaljusticationforthisimposedcompromise.Second,suchaschemeleadstoaconservativealgorithm,astheacceptanceprocedureresultsinareductionofthepopulationvariabilityevenwhenthevarianceoftheselectedpointdistributionisgreaterthanthatofthecandidatepointdistribution,asillustratedinFigure 4{4 .TheresultsofempiricaltestsareshowninAppendix C foralaminateoptimizationproblem. Figure4{4:Evenwhenthevarianceoftheselectedpointdistributionisgreaterthanthatofthecandidatepointdistribution,theprobabilisticacceptanceschemeresultsinareductionofthevariance.

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85 TargetpointapproachThisstrategyisbasedonthehypothesisthatthedistributionofselectedpointsisbestrepresentedintheauxiliaryvariablespace.Tothisaim,asetoftargetpointsaresampledfrompV,thenthenewpopulationisobtainedbyretainingthecandidatepointsthatareclosesttothesetargetpoints 4 .Dependingonthedegreeofcondenceplacedinthedistributionsp1;:::;nandpV,theirrelativeweightinthecreationofnewpointscanbeadjustedthroughtheratio=:when==1,thelaminationparameterdistributionpVplaysnoroleinthesearch;when=!1,theoptimizationisbasedprimarilyoninformationinthelaminationparameterspace.Incontrasttothepreviousapproach,whichtendedtoreducethevarianceofthesearchdistribution,thetargetpointstrategyallowstheauxiliaryvariabledistributiontocorrecttheunivariatedistributioneitherbydecreasingthevariance,orbyincreas-ingit,asshownschematicallyinFigure 4{5 .Ifthevariabilityofthepopulationgeneratedfromtheunivariatedistributionsisgreaterthanthatofthetargetdistri-butionassumedmoreaccurate,targetpointswillproportionatelypickoutmorepointsincentralregionsthaninthetails,therebyreducingthedistributionvarianceFigurea.Conversely,thevariancecanbeincreasedbyfavoringperipheralregionsifthepointsgeneratedbytheunivariatedistributionaretoofocusedaroundthemeanFigureb. 4Eachtargetpointisconsideredinturn,andthecandidatewhichhasthesmallestEuclideandistancetoitintheV-spaceiscopiedinthenewpopulationandremovedfromthepool,therebypreventingmultiplecopiesofapointfromappearinginthenewpopulation.

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86 aDecreaseinthevariance bIncreaseinthevarianceFigure4{5:Inuenceofthetwo-distributionschemeonthevariance.Thevariabilityofthesearchdistributioncanbeadjustedbasedontheauxiliaryvariabledistributioninarealsituation,themeansofthetargetandcandidatedistributionsaredierent,butthesamemechanismsoperate.

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87 4.3Thedouble-distributionoptimizationalgorithm4.3.1GeneralalgorithmTherepresentationofthedistributionofselectedpointsprovidedbythetargetpointprocedurewasadoptedherebecauseithasaclearerinterpretation,andallowstherelativecontributionofeachofthedistributionstobeadjusteddependingonthecondenceplacedineachmodel.Theresultingestimationofdistributionalgorithm,namedDouble-DistributionOptimizationAlgorithmDDOAisshowninFigure 4{6 Figure4{6:Flowchartofthedouble-distributionoptimizationalgorithm. ThemainstepsofthealgorithmareidenticaltothegeneralEDApresentedinSection 2.3 ,butthegeneralformofpxissubstitutedwiththetwo-distributionstatisticalmodelconsistingoftheunivariatedistributionsofthedesignvariables

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88 pxk,andtheauxiliaryvariabledistributionpV.Attheinitializationandestimationsteps,thetwodistributionsaretreatedinparallel,andthegenerationofnewpointsatthesamplingstepisperformedaccordingtotheprocedurepresentedinSection 4.2.2 .ThespecicformofthedistributionpV,aswellastheproceduresimplementedtoestimateitandtosamplepointsfromitwillbepresentedinthenextsection.AsinUMDA,twoalternativediversitypreservingmechanismswereaddedtothealgorithmtopreventprematureconvergencetoalocaloptimum:aconventionalmutationoperator,appliedwithprobabilitypmpervariableafterthesamplingstep,andalowerboundonthemarginalprobabilitiescf.Section 3.3 ,whichensuresthateachpointofthedesignspacehasanon-zeroprobabilityofbeingvisitedatanytimeofthesearch.4.4ApplicationtocompositeoptimizationThealgorithmisspecializedtocompositelaminateoptimizationthroughthechoiceofproperauxiliaryvariablesandappliedtothreeproblemspresentingdier-entfeatures:anin-planeproblem,wheretwoauxiliaryvariablescapturetheoverallresponse,anin-plane/out-of-planeproblem,wherefourauxiliaryvariablesmustbeused,andaproblemwheretheauxiliaryvariablesprovideonlypartialinformationabouttheobjectivefunction.4.4.1DDOAforcompositesThedouble-distributionoptimizationalgorithmmodelsvariableinteractionsthroughthedistributionofhigher-orderquantitiesthatcapturejointinuencesofseveralvari-ables.Inthecaseofcompositelaminates,theprimarydesignvariablesaretheber

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89 anglesk,k=1;:::;n.Auxiliaryvariablesappearnaturallyduringthehomogeniza-tionthroughthethicknessofstinessproperties:thelaminatestinessisobtainedasaproductofmaterialinvariantsandgeometriccontributions,calledlaminationparameters TsaiandPagano 1968 .Theonlynon-zeroextensionallaminationpa-rametersViandexurallaminationparametersWiofabalancedsymmetriclaminate[1;2;;n]sare,respectively,thefollowing 5 :Vf1;3g=2 hZh=20fcos2;cos4gdz=1 nnXk=1fcos2k;cos4kg.2andWf1;3g=24 h3Zh=20fcos2k;cos4kgz2dz=1 n3nXk=1akfcos2k;cos4kg;.3wherehdesignatesthetotallaminatethicknessandak=n)]TJ/F22 11.955 Tf 12.598 0 Td[(k+13)]TJ/F15 11.955 Tf 12.598 0 Td[(n)]TJ/F22 11.955 Tf 12.598 0 Td[(k3cf.Appendix A{1 formoredetails.NotethatEquations 4.2 and 4.3 deneafeasibledomainLinV1;V3;W1;W3,whichisthesetoftheimagesofallpossiblelaminates[1=:::=n]s. 5Here,theasteriskdoesnotdenotetheoptimum,butisacommonwayofdis-tinguishingthenormalizedlaminationparametersfromtheirbasicform,notusedinthiswork.

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90 Theutilityoflaminationparametersforcompositeoptimizationhasbeendemon-stratedbyseveralresearcherse.g. Miki 1986 ; TodorokiandHaftka 1998 .Perform-ingtheoptimizationinthelaminationparameterspacepresentsseveraladvantages:insomecases,itforcesconvexityoftheobjectivefunction Foldageretal. 1998 ,itreducesdimensionality,itrenderstheproblemamenabletocontinuousoptimiza-tionmethods.However,theproblemofndingthelaminatethatcorrespondstotheoptimumlaminationparametersVopt=V1opt;V3opt;W1opt;W3optisnon-trivial:itconstitutesanoptimizationprobleminitself,andcanhaveonesolution,orseveralsolutions,orevennone.Thealgorithmforlaminateoptimizationusesasimpleunivariatemodeltorep-resenttheangledistributionp1;:::;n,andaddsvariabledependenciesbybiasingthesearchbasedonthedistributionofselectedpointsinthelaminationparameterspace.Sincetheberangleskarediscrete,thelaminationparameterscanalsoas-sumeonlyanitenumberofdiscretevalues.However,thenumberofvaluesthattheycantakeincreasesrapidlywiththeproblemdimensionn.Furthermore,thesevaluesllupthewholelaminationparameterspacewhennbecomeslarge.Forthesereasons,theirdistributionisbestdescribedbyacontinuousmodel.WewillapplythegeneralDDOAalgorithmpresentedinSection 4.3.1 ,substitut-ingtheprimaryvariabledistributionspxk,k=1;:::;nwiththeberangledistri-butionspk,andthelaminationparameterdistributionpVVisthesetoflamina-tionparametersconsideredanddependsontheproblemathand:V=V1;V3forin-planeproblems,V=W1;W3forout-of-planeproblems,andV=V1;V3;W1;W3forproblemsinvolvingboththeextensionalandexuralpropertiesofthelaminate

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91 asauxiliaryvariabledistribution.4.4.2IssuesassociatedwiththerepresentationoftheauxiliaryvariabledistributionTwoaspectsofthealgorithmdeservespecialconsideration:therepresentationandtheestimationoftheprobabilitydistributionpVandthetargetpointcreationprocedure.RepresentationofpVLaminationparametersareinherentlycontinuous,thereforeacontinuousprob-abilitydensityfunctionisusedtorepresentpV.MostoftheworksoncontinuousEDAstodateuseunivariatenormaldistributionstorepresentthedistributionofselectedpoints SebagandDucoulombier 1998 ; Gallagheretal. 1999 .Thedisad-vantagesofthatmodelisthatitdoesnotmodeldependenciesbetweenvariables,andassumesaunimodalsymmetricdistribution. BosmanandThierens 2000 proposedtouseanon-parametrickerneldensityestimationKDEmethodtoachieveamoreaccurateapproximation.Forthismethodtoprovideanaccurateestimateofdistri-butions,agoodcoverageofthespaceisrequired.Sincethenumberoflaminationparametersthatweconsiderissmalltwoorfourandindependentoftheproblemdimensionn,goodspacecoveragecanbeachievedwithreasonablesamplesizes,sothatKDEisappropriateformodelingthedistributionpV.Inthekerneldensityestimationmethod,akernelKuisplacedateachsamplepoint.ThedistributionpVisobtainedaspV=1 Xi=1KV)]TJ/F37 11.955 Tf 11.955 0 Td[(Vi:.4

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92 Inthiswork,weusedGaussiankernels:Ku=1 d=2dexp)]TJ/F37 11.955 Tf 10.494 8.088 Td[(uTu 2.5wheredisthedimensionofuandthevariance2isthebandwidththatneedstobeadjusted:asmallvalueofincreasestheresolutionbutalsoincreasesthevarianceoftheestimatewhenfewdatapointsareavailable.Severalmethodsforadjustingthevalueofexist,suchastrial-and-error,maximumlikelihood,oradaptivestrategy.Inthiswork,amaximumlikelihoodmethodwasusedseeAppendix B foradescriptionofthemethod,anddetailsabouttheprocedure.TargetpointcreationprocedureTheuseofunbounded-supportdensityfunctions,suchasnormalkernels,fortherepresentationofpVcomplicatesthesamplingofpointsintheboundedlaminationparameterspace,becauseinfeasibletargetpointspointsthatdonotlieinthefeasibledomainL,seeAppendix A canbegenerated.Twostrategieswereconsidered: onecaneitherforcethetargetpointstolieinthefeasibledomainbysamplingnewpointsuntilfeasiblepointsareobtained, orallowinfeasibletargetpointstobecreated,consideringthatallthepointsofthenewgenerationwillbefeasiblebyconstructionthepointsarepreviouslycreatedinthe-space,cf.Figure 4{6 .Theformerapproachhastwodisadvantages:rst,theexactboundariesofthefeasi-bledomainareonlyknownforafewsimplecombinationsoflaminationparametersV1;V3andW1;W3.Forgeneralcombinations,approximaterelationsonlyare

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93 available Diaconuetal. 2002a .Second,testsshowedthattheperformanceofalgo-rithmsbasedontherejectionofinfeasibletargetpointsdeteriorateswhentheoptimalieclosetotheboundarycf.Appendix B .Consequently,thelatterschemeonlywasusedinthiswork.4.4.32DdidacticexampleThevalidityoftheapproachisnowdemonstratedonthe2Dtnesslandscapepre-sentedinSection 4.1 .WesawthattheprobabilitydistributionyieldedbyaunivariatemodelapproximatedthedistributionofselectedpointsreproducedinFigureaverypoorly.Figurebshowsthedistributionobtainedbythedouble-distributionprocedurewithlinearrankingselection,=30,=30,=300,averagedover100independentruns.Clearly,usingtheinformationaboutthedistributionofauxiliary aJointdistributionofselectedpoints bDistributiongeneratedbyDDOAFigure4{7:Eectofthetwo-distributionsamplingprocedure.Theauxiliary-variable-basedacceptanceschemeintroducesvariabledependenciescomparedtoFigure 4{1 variablesdramaticallyimprovestheaccuracyofthesearchdistribution:thehigh-probablityareaaround5,55causedbyunivariatesamplingisstillobservable,but

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94 itsrelativeweightinthedistributionisconsiderablyreduced.Onthecontrary,theshapeofthevalley,whichwashardlyvisibleintheunivariatedistributionappearsclearlyinthedistributionproducedbyDDOA.4.4.4ExtensionalproblemTherstoptimizationproblemwasmaximizingthetransversein-planestinesscoecientA22subjecttoaconstraintontheeectivePoisson'sratioe:maximizeA22=hU1)]TJ/F22 11.955 Tf 11.956 0 Td[(U2V1+U3V3suchthatleu.6TheeectivePoisson'sratioeisafunctionofV1andV3:e=A12 A22=U4)]TJ/F22 11.955 Tf 11.956 0 Td[(U3V3 1)]TJ/F22 11.955 Tf 11.956 0 Td[(U2V1+U3V3;.7andU1,U2,U3,U4arematerialinvariants,obtainedfromthematerialpropertiesshowninTable 3{3 p. 68 ,andthetotallaminatethickness 6 wasxedath=0:2in.ThelowerandupperlimitsofthePoisson'sratiowerel=0:48andu=0:52.Theinequalityconstraintswereenforcedbyapenaltyapproach:F=8>>><>>>:A22ifg0A22+pgifg<0.8 6Inpractice,theplythicknessisxed,andthetotalthicknessdependsonthenumberofplies.However,weusedaxedtotalthicknessinthisworktoobservetheeectofdimensionalityforagivenobjectivefunctiontheordersofmagnitudeofA22anderemainthesameforaxedh.

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95 whereg=mine l)]TJ/F15 11.955 Tf 11.955 0 Td[(1;1)]TJ/F22 11.955 Tf 13.15 8.088 Td[(e u:.9Thevalueofthepenaltyparameterpwassetto5:0106soastoguaranteeafeasiblesolution.Theberorientationswerechosenfromf0;22:5;45;67:5;90g.Thetnesslandscapeforn=2isplottedinFigure 4{8 :thefunctionpresentstwonearlycircularhigh-tnessregions,correspondingtothefeasibledomain.Themaxi-mumislocatedintheouterregion,whiletheinnerhigh-tnessregionisonlylocallymaximum.Thecurvedshapeoftheseareaswillgiverisetovariabledependenciesinselectedpoints,hencemakingunivariatestatisticalmodelinaccurateinthesemodelsmaximumprobabilityareasarealwaysalignedwiththeaxes,cf.Section 4.1 Figure4{8:FitnesslandscapeofthepenalizedA22forn=2. ThisproblemisanidealcasefortheDDOAalgorithmbecausetheobjective

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96 functioncanbeexpressedentirelyintermsofV1andV3,sothatthedistributionofpromisingpointscanbecharacterizedcompletelyintheV1;V3-plane.Inthisproblem,variableinteractionsoriginatefromtwofactors.First,theconstraintcreatesanarrowridge"inthedesignspaceandgivesrisetovariabledependenciessimilartothoseobservedinFigure 4{1 .Second,sincetheorderofthepliesisirrelevant,severalcongurationsoftheoptimallaminatemaybepresentinthesetofselectedpoints,hencecreatingadditionalvariabledependencies 7 .Twocasesweretested:n=6andn=12.Intheformercase,theoptimumstackingsequencewas[454=67:52]soranypermutationofthesameangles,withatransversestinessA22=1:897106lb/in,andaneectivePoisson'sratioe=0:519.Inthecasen=12,theoptimumstackingsequencewas[459=67:5=904]soranypermutationofthesameangles,withatransversestinessA22=1:922106lb/in,andaneectivePoisson'sratio 8 e=0:489.TheDDOAalgorithmwascomparedtotwoevolutionaryalgorithms:astan-dardgeneticalgorithmpopulationsizeof30,two-pointcrossoverwithprobabilitypc=1:0,linearrankingparentselectionandtheunivariatemarginaldistributionalgorithm MuhlenbeinandMahnig 2000 withlinearrankingselection,population 7Forinstance,iftheoptimumlaminateis[1=2]s=[30=45]s,[45=30]sisalsoasolution.Ifweknowthat1=30,theprobabilityof2=45beingoptimumdenotedbyp2=45j1=30is1andtheprobabilityof2=30beingoptimumis0:clearly,p2j1isafunctionof1,whichisthedenitionofvariabledependence.8Theconstraintisnotstrictlyactivebecauseofthediscretenessofthevariables.

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97 size=30andselectedpopulationsize=30duplicateswereallowedinthese-lectedpopulation.ThesesettingswerealsousedforDDOA,andacandidatepoolsize=150waschosen.Inthethreealgorithms,mutationconsistedinchangingthevalueofavariabletooneofthetwonearestvalues,andaxedmutationratepm=0:02pervariablewasused.Naturalcodingwasemployedforthethreeal-gorithmsnotransformationwasappliedtothevariablesbeforeusingtheminthealgorithms.Noelitiststrategywasimplemented,toisolatetheeectoftheproba-bilisticmodel 9 .Thebandwidthwaschosenbyamaximumlikelihoodmethod,whichprovidesanupperboundfortheoptimalbandwidthbydeterminingthevalueofthatbestapproximatesthedistributionofselectedpointsattherstgenerationofthealgorithmseedetailsinAppendix B .Usingthisapproachweobtained=0:2forn=6and=0:15forn=12.Theperformanceoftheoptimizationwasassessedbyestimatingthemeanob-jectivefunctionevaluationofthebestpointateachgeneration,andthereliabilityR,estimatedover50independentruns.Figure 4{9 comparesthethreealgorithmsforn=6.DDOAisparticularlye-cientinbeginningofthesearch:asearlyastherstgeneration,itsmeanobjectivefunctionissignicantlyhigherthanthatofthetwootheralgorithms.Thisempha-sizestheimportanceofinitialization:byimposingauniforminitialdistributioninthelaminationparameterspace,DDOAreducesthebiastowardlaminatesmadeof 9Howeverthebestsolutionfoundateachgenerationisrecorded,sothateventhoughitdoesnotinuencethesearch,itisnotlostfortheuser.

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98 multipleangleswhentheplyorderisirrelevant,laminatessuchas[02=45=902]scanappearundersixpermutations,whereas[06]shasonlyoneconguration,henceitislesslikelytoappearbyuniformsamplingintheanglespace.DDOA'shighermeanobjectivefunctiontranslatesintoahigherreliability,indicatingthatthealgorithmnotonlyconvergestohighobjectivefunctionregionsfasterthanitscompetitors,butyieldsthetrueoptimummoreconsistently. a bFigure4{9:MeanmaximumtnessandreliabilityfortheconstrainedmaxA22prob-lem,n=6runs. Theperformanceimprovementcanbeattributedtothemoreecientinitializa-tion,andtoamoreaccurateestimationofthedistributionofselectedpointsachievedbyincorporatingvariabledependenciesviathelaminationparameters.WorkingintheV1;V3-planewasespeciallybenecialforthisproblemwheretheplyorderisirrelevantbecauseallthecongurationsofagivenlaminategetmappedintoasinglepointintheV1;V3-plane,sothatthedistributionbecomesverysimpleinthatspace,whileaunivariatemodelinthe-spacedescribesthedistributiononlyimperfectly.

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99 Inthecasen=12,theconvergencecriterionwasrelaxedbyusingapractical"optimum,oracceptablesolution,denedas99%ofthemaximumobjectivefunction,insteadofthetrueoptimum.TheperformanceofthethreealgorithmsiscomparedinFigure 4{10 .Thegainachievedbyincorporatinginformationaboutthelaminationparameterdistributionisverysignicantforthisproblem.DDOA'ssuperiorityisparticularlydramaticintheinitialphaseofthesearch,fromthestartofthesearchto400evaluations,whereconvergencetowardhighobjectivefunctionregionsbenetsfromtheinformationaboutpromisingdesignsinthelaminationparameterspace. a bFigure4{10:MeanmaximumtnessandreliabilityfortheconstrainedmaxA22problem,n=120runs. ThehighercomparativeperformanceofDDOAforlargerncanbeexplainedbythefactthatDDOAperformspartofthedataanalysisintheV1;V3-plane,whosedimensionisinvariant,consequently,usingthelaminationparameterstoguidethesearchpartiallyneutralizestheeectofdimensionalityonthereliability.

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100 4.4.5Extensional-exuralproblemThesecondproblemwasminimizingtheabsolutevalueofthelongitudinalcoef-cientofthermalexpansionCTE xofasimplysupportedrectangulargraphite-epoxyplatesubjecttoalowerboundontherstnaturalfrequencyf1.Theproblemwasformulatedasfollows:minimizej xjsuchthatf1fmin.10wherethelongitudinalCTEisafunctionofV1andV3 Gurdaletal. 1998 ,p.154,seeAppendix A fordetails: x=V1K2U1)]TJ/F22 11.955 Tf 11.955 0 Td[(K1U2+K2U4)]TJ/F22 11.955 Tf 11.956 0 Td[(V12K2U2+2K1U3V3+K1U1)]TJ/F22 11.955 Tf 11.955 0 Td[(U4 22V3U3U4+U1)]TJ/F22 11.955 Tf 11.955 0 Td[(V12U22+U21)]TJ/F22 11.955 Tf 11.955 0 Td[(U24.11whereK1=U1+U41+2+U21)]TJ/F22 11.955 Tf 11.955 0 Td[(2K2=U21+2+U1+2U3)]TJ/F22 11.955 Tf 11.955 0 Td[(U41)]TJ/F22 11.955 Tf 11.955 0 Td[(2K3=U21+2+2U3+U51)]TJ/F22 11.955 Tf 11.955 0 Td[(2:TherstnaturalfrequencyisafunctionofW1andW3throughD11,D12,D22,andD66cf.Appendix A fordetails:f1=2 p hr 1 a4D11+2 a2b2D12+2D66+1 b4D22:.12Notethattheobjectivefunctioncanbeexpressedintermsofthelaminationparam-etersonly,asinthepreviousproblem.Theminimizationproblemwasrecastintoamaximizationproblembydening

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101 theobjectivefunctionasfollows:F=8>>><>>>:2)-222(j xjifg02)-222(j xj+pgifg<0.13whereg=f1 fmin)]TJ/F15 11.955 Tf 11.955 0 Td[(1:.14Forthisproblem,thematerialpropertiesshowninTable 3{3 p. 68 wereused.Thecoecientsofthermalexpansioninthelongitudinalandtransversedirectionswere1=0:0210)]TJ/F21 7.97 Tf 6.586 0 Td[(6K)]TJ/F21 7.97 Tf 6.587 0 Td[(1and2=22:5010)]TJ/F21 7.97 Tf 6.586 0 Td[(6K)]TJ/F21 7.97 Tf 6.587 0 Td[(1,respectively,andthedimensionsoftheplatewerea=30inandb=15in.Theminimumfrequencywasfmin=150Hz.Thetnesslandscapeforn=2isshowninFigure 4{11 Figure4{11:FitnesslandscapeofthepenalizedCTEproblemforn=2. Twocaseswereconsideredintheoptimization:n=6andn=12.Inthe

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102 formercase,threefeasiblelaminatesyieldedtheminimumlongitudinalCTEof x=8:5410)]TJ/F21 7.97 Tf 6.587 0 Td[(7K)]TJ/F21 7.97 Tf 6.587 0 Td[(1:[902=67:5=08]s,whichhadarstnaturalfrequencyoff1=156:25Hz,[902=02=67:5=06]s,withf1=152:53Hz,and[902=08=67:5]s,withf1=151:36Hzthesevaluesaretobecomparedtothemaximumrstnaturalfrequencyoftheplatef1=183:71Hzfor[9012]sandtheminimumCTEfortheunconstrainedproblem x=1=0:210)]TJ/F21 7.97 Tf 6.586 0 Td[(7K)]TJ/F21 7.97 Tf 6.586 0 Td[(1for[012]s.Inthecasen=12,theoptimumlaminatewas[904=67:5=06=22:56],with x=5:9510)]TJ/F21 7.97 Tf 6.586 0 Td[(7K)]TJ/F21 7.97 Tf 6.587 0 Td[(1andf1=150:13Hzthemaximumfrequencyisf1=183:71Hzfor[9024]s,andtheminimumCTEis x=)]TJ/F15 11.955 Tf 9.299 0 Td[(1:8210)]TJ/F21 7.97 Tf 6.587 0 Td[(9K)]TJ/F21 7.97 Tf 6.586 0 Td[(1for[902=452=22:53=012]s.Thepenaltyparameterwassettop=1:010)]TJ/F21 7.97 Tf 6.586 0 Td[(4.Theallowableberorientations,geneticoperators,andotherparameterswerethesameasforthein-planeproblem.Thebandwidthfoundbythemaximumlikelihoodmethodwere=0:2forn=6and=0:15forn=12.TheperformanceofthestandardGA,UMDAandDDOAisshowninFigures 4{12 and 4{13 a bFigure4{12:MeanmaximumtnessandreliabilityfortheconstrainedminCTEproblem,n=6.

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103 a bFigure4{13:MeanmaximumtnessandreliabilityfortheconstrainedminCTEproblem,n=120runs. Forthisproblem,acleardierenceinthebehaviorofthealgorithmscanbeobservedbetweenthetwocasesn=6andn=12.Intheformercase,theoptimumlaminatesdonotexploitplyinteractions:thebestlaminatesarejustobtainedbyusingasmany0-degreepliesaspossible,astheypossessthelowestlongitudinalcoecientofthermalexpansion,andcomplementingbytwostacksofpliesorientedatlargeanglesinouterlayerstosatisfythefrequencyconstraintweshallcalltheselaminatessolutionsofTypeI.Inthecasen=12,theoptimumismorecomplexbecauseitsresponseresultsfromthecombinedactionofalltheplies:the22:5stacksalonewouldyieldanegativeCTEof x=)]TJ/F15 11.955 Tf 9.299 0 Td[(3:9510)]TJ/F21 7.97 Tf 6.586 0 Td[(6K)]TJ/F21 7.97 Tf 6.586 0 Td[(1.Whenusedinassociationwith02,45,and902stacks,whichhavepositiveCTEs,theyproduceverylowCTElaminate.Becausetheyaremadeofmoderatetolargeangles,suchlaminateseasilysatisfythefrequencyconstraintweshallrefertotheselaminatesassolutionsofTypeII.The22:5stacksarenotparticularlygoodindividually:theyareonlyusefulinsofarastheycompensateforlarger-CTEplies,givingrisetostrong

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104 plyinteractions.Theperformanceofthealgorithmsreectsthisdierenceinthestructureofthesolutions:inthecasen=6,nosignicantdierencebetweenthethreealgorithmscanbeseen.Incontrast,inthecasen=12,DDOA'ssuperiorityappearsclearly,bothintheaveragebestobjectivefunctionandinthereliability.WhileGAandDDOAtendtoconvergetosub-optimalTypeIsolutions,DDOAreliablyndsTypeIIsolutionsbyidentifyingpromisinganglecombinationsthroughthelaminationparameters.4.4.6StrengthproblemThethirdproblemwasmaximizingtheloadfactors,usingtherst-ply-failurecriterionbasedonthemaximumstrain,foraglass-epoxylaminatesubjectedtothein-planeloadingNx=)]TJ/F15 11.955 Tf 9.298 0 Td[(1000103N/m,Ny=200103N/m,Nxy=400103N/m:maximizes=nmink=1minmaxt1 1k;)]TJ/F22 11.955 Tf 18.292 8.087 Td[(c1 1k;maxc2 2k;)]TJ/F22 11.955 Tf 18.292 8.087 Td[(t2 2k;ult12 j12kj.15wheretheloadfactorsisthecoecientbywhichtheloadhastobemultipliedforthestructuretofail.ThematerialpropertiesusedforthisproblemareshowninTable 3{4 p. 74 .Thetotalthicknessofthelaminatewash=2cm.Theparticularityofthisproblemisthatthelaminationparametersprovideonlypartialinformationabouttheobjectivefunction:s=sV1;V3;1;:::;n:Weappliedthethreealgorithmstothecasen=12.Theoptimallaminatefor

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105 thatproblemwas[014=67:55]soranypermutationofthesameangles,whichyieldedaloadfactors=4:74.Theparametersettingswerekeptunchangedfromthepreviousproblems.Abandwidthof=0:15wasusedforDDOA.Themeanbestobjectivefunctionandthereliabilityforapracticaloptimumof95%oftheoptimumarepresentedinFigure 4{14 .Thealgorithms'performanceisdirectlyinuencedbytheamountofvariableinteractionsusedtoguidethesearch:UMDA,whichtreatsallthevariablesinde-pendentlyquicklyconvergestoanaveragebestobjectivefunctionofs3:6in400evaluationsandstagnatesinthissub-optimalregionafterward;two-point-crossover-basedGApreservessomelinkagebetweengenesandreachesahigheraveragebestobjectivefunctionofs3:85.DDOA,whichexplicitlyhandleslaminate-levelvari-abledependenciesthroughthelaminationparameters,achievesasubstantiallyhigheraverageobjectivefunctionofs4:1.DDOA'sadvantageisevenmoreremarkablewhenwelookatthereliability:thealgorithm'sprobabilityofndingasolutionwithin5%oftheglobaloptimumreaches62%for1,500functionevaluations,whileGAandUMDAstagnateat12%and14%respectively.Incontrasttothetwopreviousproblems,wherevariabledependenciesoriginatedfromconstraintsatisfactionconsiderations,thevariablesofthisproblemarerelatedwithstrongdependenciesbecausechangingtheorientationofaparticularplycausesaloadredistribution,whichdirectlyaectstheoptimalorientationofotherplies.Byallowinggroupsofvariablestobetreatedtogether,GA'stwo-pointcrossoverchangesthevalueofsomevariables,whilekeepingtheothersconstant,itsearchesthebestori-entationoftheseplieswithoutcausingsignicantloadredistribution.DDOAprovides

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106 amoreexplicitmechanismforseparatingoverallresponseandlocalplyadjustmentbyallowingmultiplelaminatesthathavesimilarstinesspropertieshencesimilarstrainstobegenerated.UMDA,whichextrapolatespromisingregionsbasedoninsucientply-levelinformation,isnotabletoreliablyproducehigh-strengthlami-nates,henceitspooraveragebestfunctionevaluation,andtheabsenceofprogress. a bFigure4{14:Meanbesttnessandreliabilityforthestrengthproblem,n=120runs 4.4.7AcomparisonofdiversityinUMDAandDDOAThediversityofthepopulationsgeneratedduringthesearchisoneoftheimpor-tantfactorsthatdeterminetheeciency:excessiveentropyinthesearchdistributionhamperstheconvergencetohigh-tnessregions,whileinsucientdiversityleadstoalocalizedsearchthatfailstoextractinformationaboutgloballyoptimalregionsprematureconvergence"inEAs.Inaddition,insomesituations,populationdiversitycanbeadesirableattributeinitself.Forinstance,theusermaybeinterestedinhavingtheoptiontochoose

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107 betweenseveralalternativesolutionstoaproblemsometimesitgiveshimachancetoaccountforfactorsthatwereleftoutintheproblemdenition.Likewise,inconstraintsatisfactionproblems,whereoneisconcernedwiththeidenticationofwholeregionsofthespace,maintainingahighlevelofdiversityisessential.Theissueofthepreservationofdiversityisanessentialbutoftenignoredaspectofestimationofdistributionalgorithms.Indeed,whenoneappliesanEDAaspresentedinSection 2 topracticalproblems,oneimmediatelyfacestheproblemofprematureconvergence"wellknowningeneticalgorithms:afterafewiterations,theprobabilityofwholeregionsofthesearchspacevanishes,consequentlytheseareasareexcludedfromthesearchdomainforthesubsequentiterations.Iftheremainingregionscontaintheoptimum,thisphenomenonisbenecial,however,thisisnotgenerallythecase.If,atsomepointoftheoptimization,theprobabilityoftheoptimumvanishes,abasicEDAwillneverbeabletondit.Twofactorscontributetothelossofoptimumfromthedistribution: 1. tness-basedselectionaimsatidentifyinggoodsub-solutionsvariablesorgroupsofvariablesthatmakeuptheoptimum.Thefrequencyofthesesub-solutionsdependsontheirmeantnessinthepopulation.Duetotheinuenceoftheothervariables,thismeantnessisarandomvariable.Dependingonitsvari-ability,thediscriminationofsub-solutionsmaybedicult,andthereisachancethatnoinstanceoftheoptimalvaluesofthevariablesconsideredwillbepresentintheselectedpoints,whichimmediatelyleadstoazeroprobabilityfortheop-timum; 2. theeectofnoisydiscriminationiscompoundedbythefactthateveninthe

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108 absenceoftness-basedselection,iterativelysamplingfromadiscretedistribu-tionandestimatingtheupdateddistributionfromthatsampleresultsinthedegeneracyofthedistributiontoasinglepoint.Evenifoptimalvaluesofallthevariablesarerepresentedintheselectedpopulation,thereisaniteprobabilitythattheywillbelostinthesamplingprocess.Thepopulationdiversitycanbemonitoredontwolevels 10 :atthevariablelevel,thediversityinthedesignvariablespaceisanindicatoroftheportionofthespacecoveredbythesearchdistribution;atthetnesslevel,thediversitycanbeusedasanindirectmeasureoftheentropyofthesearchdistribution,althoughalowtnessdiversitydoesnotimplyalowentropyofthesearchdistributionifmanydierentdesignsgetmappedintoasinglepointinthetnessspace,alargepopulationdiversityinthedesignvariabledomaincanproducealowdiversityinthecriterionspace.Whilethediversityinthedesignvariabledomainisamoremeaningfulindicatorofthestrengthofexploration,itismorediculttodene,andcomputationallymoreexpensivetoevaluatethanthediversityinthecriterionspace.DiversityinthecriterionspaceDcritwasdirectlymeasuredbythestandarddeviationofthetnessinthepopulation.Diversityinthevariablespacewasmeasuredbytheaveragepairwisecity-block"distanceinthepopulation:Dvar=2 )]TJ/F15 11.955 Tf 11.956 0 Td[(1)]TJ/F21 7.97 Tf 6.586 0 Td[(1Xi=1Xj=i+1dij;.16 10Inevolutionarycomputation,theseareoftenreferredtoasgenotypicalandphe-notypicaldiversity.

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109 wheredij=1 nnXk=1jxik)]TJ/F22 11.955 Tf 11.955 0 Td[(xjkj:.17TheevolutionofDcritandDvarwasmonitoredduringthemaximizationofthelaminatestrengthofSection 4.4.6 .Figures 4{15 and 4{16 showthecriteriondomainandvariabledomaindiversities,respectivelyforthewholepopulationandthetopthirdofthepopulation,whichapproximatelyconstitutesthebasisfortheconstruc-tionofthestatisticalmodel.WhilethepopulationstandarddeviationofUMDA Figure4{15:DiversityinthecriteriondomainforUMDAandDDOA. dropsveryrapidly,DDOAmaintainsanalmostconstantlevelofdiversitythroughoutthesearch.Maintainingahighleveloftnessdiversityisneithergoodnorbadapri-ori,asrandomexplorationexhibitssuchvariability,butoftenwastesmanyfunctionevaluationsonpoorregions:thisinformationhastobeexaminedjointlywiththemeantness,shownearlier.Forthisproblem,DDOAmanagestokeepasustainedlevelofvariability,whiledisplayingasubstantiallyhighermeanbesttnessthan

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110 Figure4{16:DiversityinthevariabledomainforUMDAandDDOA. UMDA.ThisindicatesthatDDOAnotonlymaintainsvariability,butitfocusesitpreferentiallyontterregions.VariabledomaindiversityprovidesadierentinsightintothesearchmechanismsatworkinUMDAandDDOA:clearly,thediversityofUMDAquicklyvanishes,unlikethatofDDOA,whichreachesaplateau,afteraninitialdropintherstiterations.Thelackofdiversityespeciallyintheupperpartofthepopulationmakesfurtherexplorationdicult,asthestatisticalmodelwillonlygeneratemanyduplicatesofasinglesolution.Incontrast,DDOAdoesnotsuerfromthisshortcoming:itguaranteesaminimumlevelofdiversity,thusenablingfurtherprogress.AmoredetailedpictureofthesearchdistributionsgeneratedbyUMDAandDDOAisprovidedinFigure 4{17 ,whichshowstheevolutionofthetnessdistribu-tionsofthebestsolutionfoundateachiterationoftheoptimization.InthecaseofUMDA,Figureaclearlyshowsaconcentrationofpointsarounds3:1,whichdo

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111 notcorrespondtoanymaximum,andtwohighdensityregionsaroundthelocalmax-imaL1=[010=454=67:5=904]ss=4:32andL2=[08=455=906]ss=3:89.Theoptimums=4:74doesnotappearinthedistribution.IncontrastwithDDOA,theweightofthegoodrandomsolutions"s3:1thatwerepresentinUMDAquicklyvanishes.Instead,thealgorithmfocusesonthehigh-tnesslocaloptimumL1andtheglobaloptimum. aUMDA bDDOAFigure4{17:DistributionofthetnessofthebestsolutionateachiterationforUMDAandDDOA. Thisdiversitypreservingeectisthedirectmanifestationofthediversitycom-pensatingeectmentionedinSection 4.4.2 :thedouble-distributionalgorithmanditstargetpointapprozchprovideaconvenientwayofcontrollingthevariabilityofthesearchdistributionviathevalueofthebandwidthandofthecandidatepoolsize.DiversityinjectionmechanismIntheoreticalEDAs,nodiversitypreservationmechanismisimplemented,be-causeinnitepopulationsareassumed,soeverypointinparticulartheoptimum

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112 hasanon-zeroprobabilityofbeingvisitedatanytimeofthesearch.However,prac-ticalimplementationsemploynitepopulations,anditbecomesessentialtoguardagainstlossofdiversity.AsecondreasonforaddingdiversityinjectionmechanismstoEDAsistocom-pensateforaninaccuratestatisticalmodel:indeed,theunderstandingofdiversitypreservationmechanismsasmeremeanstoallowapureEDAsearchproceduretoconvergewhendistributionsareestimatedfromnitesamplesassumesthattheEDAconvergesintherstplace.Thissupposesthatanaccuratestatisticalmodelisused,andthatjudiciousexploitationoftheinformationsummarizedinthemodelleadstotheoptimum.Often,asimpliedstatisticalmodelisused,andexploitationonlycannotyieldtheoptimum.Inthosesituations,anexploratorycomponenthastobeaddedintheformofaperturbation:diversityinjectionmechanismsplaysucharole.Finally,evenwhenanaccuratemodelisused,acombinationofexploitativeandexploratorysearchcomponentsmayturnouttobethemosteectivestrategy.Inthiswork,weprovidedtwodierentdiversitypreservingmechanismscf.Section 3.3 : Mutation: aperturbationisappliedwithprobabilitypmtoeachvariablekofeachofthecreatedpoints.Theperturbationconsistsinchangingthevalueofktooneoftheneighboringvalueswithequalprobabilitye.g.45canbechangedto22.5or67.5. Boundsonthesearchdistribution: thatdoesnotallowthemarginalprobabili-tiespkthreshold.Mutationappliestoallthevariablesandisindependentofthestateofthe

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113 searchdistribution.Theboundonmarginaldistributionsdirectlyaectsthesearchdistribution,butitsactionismorelocalized,asitonlyappliestovariablesoflowentropy.Theinuenceofthetwodiversitypreservationoperatorswasinvestigatedonthestrengthmaximizationproblem,forn=12.Thereliabilityforapracticaloptimumof95%oftheoptimumofUMDAandDDOAforvaluesofpmrangingbetween0and0.05withoutlimitationontheprobabilitydistributionisshowninFigure 4{18 .Whennomutationisusedpm=0,UMDAneverndstheoptimumbecause a b Figure4{18:EectofmutationforUMDAaandDDOAb. thedistributionquicklydegenerates,whichpreventsanyfurtherprogress.DDOAreachesaslightlyhigherreliabilityof12%.Whenthemutationrateisincreased,theeectivenessofbothalgorithmsimproves,thoughUMDA'sreliabilityneverexceeds10%forpm=0:04.Withthesamemutationrate,DDOAreachesareliabilityof74%.Furtherincreasingtheamountofrandomperturbationcausesbothalgorithm'sperformancetodeteriorate.Theinuenceofrepairing"degenerateprobabilitydistributionsbyimposing

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114 lowerboundsonthepk'sisshowninFigure 4{19 forUMDAaandDDOAb,whichpresentsthealgorithms'reliabilityforsevenvaluesof,rangingfrom0to0.08whichrepresents2/5oftheuniformdistributionpk=cl=1=5=0:2.This a bFigure4{19:EectofboundsontheprobabilitydistributionspkforUMDAaandDDOAb. diversitypreservationmechanismsisevenmorebenecialtothesearchperformancethanmutation:UMDA'sreliabilityreaches60%for=0:04,andDDOA'sreliabilityattheendoftheoptimizationincreasesto80%forthesamevalueof.Asinthecaseofmutation,highervaluesofcausetheperformanceofbothalgorithmstodrop.Theadvantageofboundsontheprobabilityovermutationcanbeexplainedbythefactthatmutationisacostlywayofreintroducinglostvariablevalues.Recoveringalostvaluethroughmutationmaytakealargenumberofiterationsifpmisnotsucientlyhigh,orcauselargeperturbationsifpmislarge.Repairingthedistributionguaranteesthatallvalueshaveachanceofbeinggenerated,whileinictingonlymarginalperturbationtotheconvergenceprocess.Whilebothmutationandthelowerboundonmarginalprobabilitieshavean

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115 obviouspositiveinuenceonbothalgorithms,DDOAclearlybenetsmorethanUMDAfromthisinjectionofrandomness:evenwithhighmutationrates,UMDA'sreliabilityremainspoor,andwhenlimitsontheprobabilitiesareimposed,DDOAperformsbetterthanUMDAonaverage,anditsreliabilityislesssensitivetothevalueofthanthatofUMDA.ThiscanbeexplainedbyobservingthatthemethodusedbyDDOAtogeneratenewpointsallowsittoeithercompensateforalackofdiversityifthepopulationbecomestoouniform,orreduceitsdiversitywhenheavyperturbationshavebeenintroducedcf.Section 4.2.2 .IntheGaussiankernelmodeloftheauxiliaryvariabledistribution,thebandwidthdeterminestheminimumamountofdiversityinthelaminationparameterspace,hencetheintrinsicexplorationinthealgorithmwithoutadditionalinjectionofdiversity.4.5PerformancewithoptimizedparametersSofar,theemphasiswasplacedonstudyingtheinuenceofasmallnumberofparameters.Inthissection,aparameterstudyisperformedforeachofthethreealgorithmsGA,UMDA,andDDOA,andtheoptimizedalgorithmsarecompared.4.5.1ParameterstudyandbestsettingThetestproblemisthestrengthmaximizationproblempresentedinSection 4.4.6 ,withn=12.ThecharacteristicsofthealgorithmsarepresentedinTable 4{1 .Foreachconguration,thealgorithmswererun50times,andthereliabilitywascomputedateachiteration.Foreachofthethreealgorithms,thebestsettingwasdeterminedbasedonthereliabilityattheendofthesearch,setto3,000functionevaluations.Theoverallbest

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116 Table4{1:Characteristicsofthethreealgorithms GA SelectionLinearranking Childpopulationsizef12;15;20;40;60;80;100g Parentpopulationsize= Mutationratepmf0:005;0:01;0:02;0:03g UMDA SelectionLinearranking Populationsizef12;15;20;40;60;80;100g Selectedpopulationsize= Probabilitylimitf0;0:01;0:02;0:04;0:06g DDOA SelectionLinearranking Populationsizef12;15;20;40;60;80;100g Selectedpopulationsize= Mutationratepmf0:005;0:01;0:02;0:03g Bandwidthf0.05,0.1,0.2,0.3g Candidatepoolsizef2;5;20g Probabilitylimitf0;0:01;0:02;0:04;0:06g congurationswereobtainedforthatcriterionwerethefollowing: GA: ==80,pm=0:02; UMDA: ==40,=0:06; DDOA: ==40,=0:1,=200,=0:06.NotethatGArequiresasubstantiallylargerpopulationthanthetwoEDAs.InthecaseofDDOA,thebestperformanceisnotachievedforthelargervalueofthepoolsize,butfor=5:thiscanbeexplainedbythefactthattheauxiliaryvariablescaptureonlypartoftheoverallresponse,hencetheirdistributionisnotsucienttoguidethesearch.ThereliabilityoftheoptimizedalgorithmsisshowninFigure 4{20 .Thesamehierarchyasintheprevioussectionwherereasonable"valuesoftheparameterswereused,withoutextensiveparameterstudyisobserved:DDOAremainsthemostecientalgorithm,althoughitssuperiorityissignicantlyreducedtobecomparedtoFigure 4{14 .GAbenetsfromalargerpopulation,whileplacingboundson

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117 Figure4{20:ComparedperformancesoftheoptimizedGA,UMDA,andDDOA. UMDA'smarginaldistributionsdrasticallyimprovesitsbehavior,asseeninSection 4.4.7 .Anotherinterestingresultoftheparameterstudyconcernsthesensitivityofthealgorithms'performancetothevalueoftheparameters:inarealisticsetting,onedoesnothavetheluxurytoperformmanytestsinordertoobtainthebestalgorithmparametervalues.Itisthereforeimportanttoexaminehowchangesintheparametersaecttheperformance.Figure 4{21 showsthereliabilityofthethreealgorithmsforsevendierentpopulationssizes,theotherparametersbeingsettotheirbestfoundvalue.Thegraphsrevealsastrikingdierenceinthealgorithms'sensitivitytochangesinthepopulationsize:whilethetwoEDAsdisplayarelativestabilitytothechoiceof,theGA'sreliabilityisstronglydependentonitsvalue.Inparticular,lowervaluesofyieldverypoorreliability:unlikeUMDAandDDOA,whichrelyonanexplicitlymechanismthatpreventsthedistributionsfromconvergingtheboundsonmarginal

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118 aGA bUMDA cDDOA Figure4{21:Bestschemesforsevenpopulationsizes,forGA,UMDA,andDDOA. distributions,GAmustresorttothemoredisruptivemutationoperator,oruselargepopulations.Tosummarize,eventhoughGAandUMDAcomparewellwithDDOAattheirbestsettings,departurefromtheseoptimalparametervaluespopulationsizeforGA,diversitypreservingmechanismforUMDAmaycauseaseveredropintheperformance,whileDDOAconsistentlyprovidesareliablesolution.

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119 4.6GeneralizationofDDOAtocontinuousoptimizationproblemTheDouble-Distributionstrategyforincorporatingstatisticalvariabledepen-denciesintothemodelofpromisingregionsisgeneralandcanbeappliedwheneverauxiliaryvariablesareavailable.Thegoalofthissectionistodemonstratethegen-eralityofthemethodbyapplyingittocontinuousproblems.4.6.1ProblemdescriptionWeconsidertheproblemofmaximizingthedistancetoapointx0:maximized=vuut nXi=1xi)]TJ/F22 11.955 Tf 11.955 0 Td[(x0i2suchthatg1x=1)]TJ/F22 11.955 Tf 14.855 8.088 Td[(r2 R20andg2x=r2 R+2)]TJ/F15 11.955 Tf 11.956 0 Td[(10;.18withx2[0;10]n,R=1,andr2=Pni=1x2i.Thereferencepointx0wasdenedinsuchawaythatoneoftheverticesofthefeasiblespacexvwastheuniqueglobaloptimum:x0=xm+xm)]TJ/F37 11.955 Tf 11.955 0 Td[(xv;.19wherexmisthemiddleofthediagonaloftheunitcubexm=1 2p ne1+e2++en.Inourproblem,wechosexv=+e1,=0:01,=R=2.Theconstraintwasenforcedbyapenaltyapproach:F=8>>><>>>:Fifmaxg1;g20F)]TJ/F22 11.955 Tf 11.955 0 Td[(pmaxg1;g2otherwise:.20Weusedapenaltyparameterp=50.Theconstraintscreateanarrowcircularridgeatr=1:005.Figure 4{22 showthecontoursofthepenalizedfunctionforn=2.The

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120 aEntiredomain bZoomonthefeasibledomainFigure4{22:Contoursofthepenalizedtnessfunctionforn=2.Thereferencepointx0=:02;0:53ismarkedbya`x'. topologyofthisproblempresentssimilarfeaturestothedidacticexampleofSection 4.1 :duetothecurvatureofthehightnessregions,andtheparticularlocationoftheoptimumatacornerofthefeasibledomain,themostprobablepointyieldedbyaunivariatemodelwillbeaninfeasibledesign,farfromthemaximumofthetnessfunction.Notingthatthisparticulartopologyiscausedbythepresenceofconstraints, 11 wedecidedtousedg1asauxiliaryvariable 12 .4.6.2ThealgorithmsContinuousversionsofUMDAandDDOA,respectivelycalledcUMDAandcD-DOA,wereimplemented.IncUMDA,thecontinuousprobabilitydensityofselected 11Thisisacommonsituationinoptimization:theoptimatypicallylieontheboundaryofthefeasibledomain,sothatthetopologyofthetnessaroundtheoptimumisforthemostpartdeterminedbytheconstraints.12Choosingg2wouldyieldidenticalresults.

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121 pointswasrepresentedbyaproductofmarginaldensities:px1;:::;xn=nYk=1pxk;.21wherethemarginaldensitieswereapproximatedbyunivariatenormaldistributionswhosemeankandstandarddeviationkwereestimatedfromthepopulationofselectedpoints.Alowerboundonminwasimposedonthestandarddeviationtopreventlossofdiversity.ThesamerepresentationwasusedforcDDOA.ThedensityoftheauxiliaryvariableVg1wasrepresentedbyanormaldistributionaswell.Thesametwo-stepalgorithmasinSection 4.3.1 wasimplementedforthecontinuouscase.ThegoalistoforcepointstolieinthefeasibledomainbylteringoutpoorcandidatesbasedonpV 13 .4.6.3ResultsanddiscussionThetwoalgorithmswereappliedtotheproblem 4.18 fortwonumbersofvari-ables,n=2andn=10.Apopulationsizeof=100waschosen,andtrun-cationselectionofratio=0:3wasapplied.ThepoolsizeusedforcDDOAwas=400points.Threedierentvaluesoftheminimumstandarddeviationweretried:min=0:001,min=0:01,andmin=0:1forthetwodistributionspxandpV.Figure 4{23 comparestheevolutionofthebesttnessforcUMDAand 13Inthisparticularcase,onemaybetemptedtodiscardallinfeasiblepoints,butitisoftenacounter-productivestrategy,asthisprecludesanyshort-cutthroughtheinfeasibledomain.

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122 cDDOA. Figure4{23:EvolutionofthebesttnessforcUMDAandcDDOA,n=2averageover50runs. Forthethreevaluesofmintested,cDDOAoutperformscUMDA.Inthecasemin=0:001,theinitialconvergenceofcUMDAisfasterthanthatofcDDOAuntilthepopulationhasfoundthetunnel:inthisinitialphase,theauxiliarydistributionhamperstheconvergencetothefeasibledomain,butoncethedistributioniscen-teredinthetunnel,itspeedsupconvergencetowardtheglobaloptimumF=1:11hencethechangeinslope,andcDDOAbecomesmoreecientthancUMDA.Thebestperformanceisachievedformin=0:01forbothalgoithms.Forthisvalueoftheminimumstandarddeviation,cDDOAneedsthreetimesfewerevaluationsthancUMDA10,000versus30,000tondtheglobaloptimum.AswasobservedforthediscreteDDOA,thetwo-distributionschemereducesthealgorithm'ssensitivitytodiversitycontrollingparameters,heremin.Thisisvisibleformin=0:1:forthatvalue,cUMDAexhibitsaslowconvergencebecause

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123 thelargevalueofminpreventsthealgorithmfromfocusingonhigh-tnessregions.Theauxiliaryvariablebasedltersubstantiallyimprovesthesearcheciencybydiscardingmeaninglesspoints.Figure 4{24 providesanotherperspectiveontheconvergenceofthedistributionsforthecontinuousdouble-distributionalgorithm.Figureashowsthevalueof1and2duringtheoptimization,andFigurebmonitorsV.Startingfromaninitialdistributioncenteredaroundp 2=2;p 2=2,themeansofthedistributionsfollowthetunnelslightlyontheoutside,asindicatedbytheauxiliaryvariabledistributionVisnegativeoutsidethecircler=1,untilitndstheoptimum,0. aConvergenceof1and2 bConvergenceofVFigure4{24:ConvergenceoftheprimaryandauxiliarydistributionsforcDDOA,n=20runs.Startingfromthecenterofthedesignspace,thedistributionsrstconvergetothetunnel",thentheyfollowitonitsoutsidedowntotheoptimum,0. Inthecasen=10,cDDOAdisplayedaverypooreciencycomparedtocUMDA,asshowninFigure 4{25 .Whilethelatterquicklyreachedhigh-tnessregions,theprogressofthedouble-distributionalgorithmwashinderedbytheauxiliaryvariabledistribution.

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124 Figure4{25:EvolutionofthebesttnessforcUMDAandcDDOAdd-cumda"ongraph,n=10averageover50runs. ThisexamplerevealsoneofEDA'slimitations,whichisenhancedbyDDOA:thedistributionpxtheoreticallyrepresentsthedistributionofpromisingregions.However,whenthespaceisonlysparselycoveredbythepopulation,entireregionsmaynotbevisitedandhencetheywillbeignoredinsubsequentiterationsofthesearchthisistheselectionerrorintroducedinSection 2.3 :EDAscanonlyapprox-imatetheselectionprobabilitypsx,sothatgooddesignsmaygetazeroselectionprobability.ThiseectisillustratedinFigure 4{26 :lowvaluesofxarenotsam-pled,consequently,theydonotcontributetotheestimateddistributionofpromisingregions.Asaresult,thesearchwillfocusonalreadyvisitedregionsandconvergencetotheoptimumwillbesloweddown.Theironicconsequenceofthiseectisthatpoorstatisticalmodels,suchascUMDA,whichinaccuratelyrepresentpromisingregionsattimet,willsamplemoreoftenfromareaswherenoobservationsofhigh-tnesspointshavebeenmade.In

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125 Figure4{26:Whenregionsofthedesignspacearenotcoveredbythepopulation,noinformationaboutthemiscollected,andtheyareexcludedfromthesearchinsubsequentiterations. particular,theywillmoreoftenbeabletocreatepointswherenoobservationatallhasbeenmade,whichhelpscUMDAescapeearlygood"butoverallpoorregions.Thisiswhatisobservedforthe10variableproblem.Incontrast,cDDOA,whichconstructsamoreaccuratemodelofpx,moreecientlyconstrainsthesearchtoalreadyvisitedareas,eventhoughtheydonotcontaintheoptimum.Specically,theauxiliaryvariableVfavorscandidatesthatarelocatedonthesamespherer=constantastheselectedpoints,whichpreventsconvergencetowardhigh-tnessregionsintheearlystagesoftheoptimization,whenthepopulationisfarfromthefeasibledomain.4.6.4ImprovementtothealgorithmOnewayofaddressingtheproblemidentiedintheprevioussectionistodras-ticallyincreasethepopulationsoastoensureagoodcoverageoftheentiredesignspace.However,thisisnotapracticalsituationinhigh-dimensionalspaces,asthenumberofpointsrequiredwouldbeprohibitive.Anewstrategywasproposedtoaddressthecasewhereinsucientinformationisavailabletobuildasatisfactory

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126 statisticalmodelofpromisingregions.Theideaistodetectsituationsinwhichthestatisticalmodelcannotbetrusted.Onesuchsituationoccurswhentheselectedandnon-selectedpointsarelinearlyseparableinagivenspace.Insuchacase,theopti-mumislikelytolieoutsidetheregionmarkedbytheselectedpointsinthedirectionperpendiculartotheseparationboundarygoingfromthenon-selectedtotheselectedpoints,inasimplex-likemanner NelderandMead 1965 .Intheseidentiedpatho-logicalcases,theproposedstrategyconsistsinkeepingthecandidatepointswiththemostextremevariablevaluesinthedirectionofincreasingtness.Thedecisiontodisregardthestatisticalmodelandtobasethesearchondirectionalityinsteadismadeonthebasisofthedegreeofseparationofselectedandnon-selectedpointsintheauxiliaryvariablespace. Figure4{27:EvolutionofthebesttnessforcUMDAandcDDOAdd-cumda"ongraph,n=10averageover50runs. Thisstrategywasimplementedforthe10-variableproblembylookingatsepa-rationintheV-spacewhichiseasysinceitisunidimensionalhere.Themeanbest

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127 tnessisshowninFigure 4{27 .ThemodiedcDDOAclearlyoutperformscUMDA.ThedetectionofpathologicalcasesbasedongoodandbadpointswhenEDAsusu-allyconsideronlygoodpoints,andtheimplementationofanappropriatestrategytodealwiththesesituationsleadstoadramaticimprovementofcDDOA'sconvergencevelocity.Theconceptofalternatingbetweenprobabilisticanddirectionalsearchhasprovenitsvalidity.4.7ConclusionInthischapter,anoriginalstrategyforrepresentingvariabledependencieswasproposed:theideaistocombinetwosimplestatisticaldistributions:aunivariatedis-tributionofthedesignvariables,andanapproximatedistributionofasmallnumberofauxiliaryvariablestoobtainacomplexdistribution.Theauxiliaryvariablesrepresentjointinuencesofthedesignvariablesonthetnessfunction.Suchvariablescanbefoundinmanyoptimizationproblems.Inthischapter,thetwo-distributionalgorithmDDOAwasappliedtodiscretelaminateoptimizationproblems,andtoacontinuousoptimizationproblem.ExperimentalresultsshowedthattheproposedstrategyconsistentlyoutperformsUMDAandaGA.AnanalysisofthediversityrevealedthatDDOAmaintainsahighlevelofentropyinthesearchdistribution,andthatiteectivelyallocatesthatdiversitytohigh-tnessregions.Finally,astrategycombiningdistribution-basedanddirectionalsearchtech-niqueswasproposedtoaddressoneofEDA'slimitations,namelythedicultyto

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128 identifypromisingregions:themethoddetectspathologicalsituationswherethedis-tributionofpromisingareasisnottobetrusted,andwhereadirectionalapproachismoreappropriate.Experimentalresultsonacontinuousproblemshowthatthisstrategycandramaticallyimprovetheeciencyofthealgorithm.

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CHAPTER5CONCLUSION5.1SummaryofthendingsThegeneralizationoftheuseofcompositematerialsinrecentyearshasspurredintensiveresearchoneectiveoptimizationtechniquessuitedfortheparticularprob-lemsthatarisewhendesigningacompositestructure.Inparticular,compositelam-inateoptimizationdeterminationofthestackingsequence,ornumberofplies,plyangles,materials,etcoftenleadstodicultcombinatorialproblemswhichconven-tionaloptimizationtechniquescannotsolveeciently.Toaddressdicultiesinherenttolaminateoptimization,suchasthediscrete-nessofthedesignvariablestheplyangles,andthemultimodalityoftheobjectivefunction,newalgorithmshavebeenproposedovertheyears.Inthiswork,theimple-mentationofanewclassofalgorithms,calledestimationofdistributionalgorithmsEDAs,forlaminateoptimization,wasinvestigated.Thesealgorithmsarestochasticoptimizationmethodsthatexplorethedesignspacebasedonasearchdistribution:ateachiteration,thedistributionofpromisingpointsisestimatedfromasetofpointsselectedfortheirhighobjectivefunctionevaluation,andthatdistributionisusedtobiasthesearchtowardgoodregionsatthenextiteration. 129

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130 5.1.1Incorporationofvariabledependenciesthroughphysics-basedaux-iliaryvariablesTherstcontributionofthisworkistheuseofphysics-basedknowledgetoim-provethestatisticalmodelofpromisingregions.Theideaistoenhancetheeciencyoftheoptimizationbymodelingthedistributionofgooddesignsmoreaccurately.ThemajorityofexistingEDAsarebasedonstringentstatisticalindependenceas-sumptionsbetweenvariablesinthepopulationsofselectedpointsorresorttocom-plexstatisticalmodelsoftheseinteractions,leadingtohighcomputationalcoststoestimatethemodelparameters.Inthiswork,weproposetomodelthecomplexdis-tributionofselectedpointsbycombiningtwosimpleprobabilitydistributions:thedistributioninthespaceofthedesignvariables,representedbyaunivariatemodelnovariableinteractions,andthedistributioninthespaceofauxiliaryvariablecho-sentocapturejointactionsofthedesignvariables.Thisisawayofincorporatinginformationabouttheformofthedistributionandimprovingthemodelaccuracyaatlowcomputationalcost.Inthecaseofcompositelaminateoptimization,thedesignvariablesaretheberangles,andlaminate-levelgeometricstinessquantitiescalledlaminationpa-rametersareanaturalchoiceofauxiliaryvariables.Theresultingalgorithm,nameddouble-distributionoptimizationalgorithmDDOA,wasappliedtothreelaminateoptimizationproblems:twoproblemsonepurelyextensionalproblemanoneex-tensional/exuralproblemwheretheresponsedependsentirelyonthelaminationparameters,andoneextensionalproblemwheretheycaptureonlypartofthere-sponse.Inthethreeproblems,theimplementedstrategyoutperformedanalgorithm

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131 basedonunivariatedistributionsonlyUMDAandastandardgeneticalgorithm,inparticularinhighlyconstrainedtnesslandscapes.ExperimentsshowedthattheadvantageofDDOAincreaseswiththeproblemdimension.Eventhoughmostofthisworkconcernscompositelaminateoptimization,theoptimizationframeworkproposedcanbeappliedinotherelds.Thelastpartofthisworkdemonstratedthevalidityofthemethodforageneralcontinuousoptimizationproblem.Inthatproblem,oneoftheconstraintsplayedtheroleofauxiliaryvariable,thusdemonstratingoneofthepossibleusesofauxiliaryvariables:ensurethefeasibil-ityofthepointsgenerated.Experimentalresultsshowedthatusingthedistributionofauxiliaryvariablesimprovestheconvergencevelocity.5.1.2ControlofthediversityAsecondemphasisofthisworkconcernedtheroleofthediversityofthesearchdistributionswasinvestigated.Indeed,theamount,aswellasthenatureofvari-abilitydirectlyinuencestheeciencyofstochasticalgorithms:toolittlevariabilityresultsinalocaloptimum,whiletoomuchrandomnessinthesearchleadstowastefulfunctionevaluationsinpoorregionsthisisthewell-knownexploitation/explorationcompromise.Ananalysisoftwodiversitypreservationmechanisms,mutationandalowerboundonmarginalprobabilities,wasconductedfortwoestimationofdistributional-gorithms:aUMDAandDDOA.Themaingoalwastopreventprematureconvergenceofthesearchdistribution.Thestudydemonstratedthatsuchadiversitypreserva-tionmechanismisessentialtoguaranteeEDAs'eciency.Itappearedthatthebest

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132 strategyistoexplicitlypreventconvergenceofthesearchdistributionbyimposingboundsonthedistributionincontrasttostandardEAs,wheresuchastrategyisimpossiblebecauseitinictssmallerperturbationstothesearchcomparedtothemutationoperator.Asideeectofthedouble-distributionalgorithmwasitsstabilitywithrespecttothesettingoftheparametersgoverningthelevelofrandomnessinthesearchmu-tationrate,boundontheprobabilities,...:thetwo-distributionsearchstrategyimplementedinDDOAprovidedacompensationmechanismforinsucientorexces-sivevariability.AnanalysisofthediversityforUMDAandDDOArevealedthatthedouble-distributionapproachprovidedawayofinjectingvariabilityinhigh-tnessregionsofthecriterionspace,makingamoreecientuseofthefunctionevaluations.5.1.3Combinationofdistribution-basedanddirectionalsearchmecha-nismsFinally,ahybridsearchstrategycombiningdistribution-basedanddirectionalmechanismswasproposed.Theideaistodetectsituationswheretheprobabilisticmodelofpromisingregionsisawed,duetoaninsucientsamplingofthedesignspace,andtoswitchtoasimplex-likedirectionalsearchmechanismmoreappropriatewhenthepopulationisawayfromhigh-tnessregions.Suchpathologicalcasesoccurwhenselectedpointsandnon-selectedpointsarelinearlyseparableinagivenspace.Thedirectionperpendiculartotheboundary,goingfromthenon-selectedtotheselectedpointswassuccessfullyusedtocomplementtheEDAsearch.

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133 5.2PotentialavenuesforfutureresearchManyaspectsofthedouble-distributionoptimizationalgorithmhavenotbeenfullystudied.Amongthefactorsthatneedfurtherinvestigation,wecanmentionthefollowingissues: eectofinitialization:itiswidelyrecognizedsee,forexample, KallelandSchoenauer 1997 thatanimprovedinitializationprocedurecandramaticallyincreasetheeciencyofevolutionaryalgorithms.DDOAcanprovideacon-venientwayofassuringamoreuniformsamplingofthecriterionspacebyincreasingthepoolsizefortherstiteration; adaptiveschemeforthedeterminationofthealgorithmparameters,inpar-ticulartherelativeimportanceoftheprimaryandauxiliaryvariablesinthesearchthroughthevalueof,choosingthemoreappropriatespacefortheoptimizationastrategyakintoautomaticcodingchoice"; elaborationofauniedframeworkforstochasticanddirectionaloptimization",followingthepreliminaryworkconductedinSection 4.6.4 .Theobjectivewouldbetogivethealgorithmtheabilitytochoosethebestsearchmechanismduringtheoptimization.

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CHAPTER6RESUME6.1OptimisationdestratiescompositesLesmateriauxcompositessontaujourd'huicourammentutilisesdansdesdo-mainesaussidiversquel'aeronautique,laconstructionnavale,l'automobileoudansdesequipementsdeloisir,ouleursproprietesspeciqueseleveessonttresappreciees.Lesmateriauxcompositessedistinguentdebeaucoupd'autresmateriauxplustra-ditionnelsparlefaitqueleursproprietesmecaniquespeuvent^etreadapteesalastructuredanslaquelleilssontemployes.Ceciconstituebiens^urungrandavantage,maisrendlaconceptiondestructurespluscomplexepuisqu'ilnes'agitplusseule-mentdechoisirlemateriauetlageometriedelastructure,maisegalementdedenirl'organisationinternedumateriau.Lesstratiessontconstituesd'unempilementdecouchesplisformeesdedierentstypesderenfortsdanslescasquinousinteressentdebresaligneesselonunedirectionprivilegieedontlacohesionestassureeparuneresine.Leursproprietesmecaniquesdependentdirectementdel'orientationdesbres.L'objectifdel'optimisationestdoncdechoisirl'orientationdechaqueplidemanierearendreextremaluncertaincritere,parexempleledeplacementenunpoint,lepoids,leco^ut,lapremierefrequencepropre,etc.,toutenrespectantuncertainnom-bredecontraintestellesquedescriteresderesistanceparexemplelacontrainteouladeformationmaximale,l'energiededeformationmaximale,lafacilitedemiseenuvre.Acausedecontraintesdefabrication,l'orientationdesbresettypiquement 134

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135 limiteeaquelquesvaleursdiscretesparexemple0,45,90.Laconceptiondestructurescompositesconduitparconsequentadesproblemesd'optimisationcombi-natoire,formulesdelafaconsuivante:maximiserF1;:::;ntelquegi1;:::;n0;i=1;:::;ravecj2A;j=1;:::;n;.1ouFestlafonctionco^ut,aussiappeleetness"danslecadredel'optimisationevolutionnaire,lesgisontlescontraintes,etAestl'ensembledesvaleurspossiblespourlesvariablesdeconceptionk,quirepresententicil'orientationdesbresdanschacundesplisdustratie.Danscetteoptique,nousnousproposonsdedevelopperdesmethodesd'optimisationdetypestatistique.6.2OptimisationstatistiqueDiversesapprochesonteteutiliseespourresoudrelesproblemesd'optimisationdestraties.Cesdixdernieresanneesontvul'emergencedemethodesstochas-tiques,aveclerecuitsimule,etplusrecemment,lesalgorithmesevolutionnaires.Lesalgorithmesevolutionnairesdontlesalgorithmesgenetiquessontunexempleconnuexplorentl'espacederechercheenassociantdesportionsdesindividuslesplusprometteursdepopulationsdepointscroisementetenappliquantdespertur-bationsaleatoiresmutation.Cesalgorithmessemontrentecacespourdenom-breuxproblemes,cependantleurecaciteoptimalen'estpasprouvee,etlaprincipalejusticationdeleurperformanceestbaseesurlametaphorebiologiquedelatheorie

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136 darwiniennedelaselectionnaturelle.LesalgorithmesaestimationdedistributionEDApourEstimationofDistribu-tionAlgorithmsconstituentuneformalisationstatistiquedesalgorithmesevolutionnaires.Ilsreprennentl'ideed'explorationpardespopulationsdepoints,maisutilisentdesmethodesstatistiquespourextraireexplicitementdel'informationsurlalocalisationdel'optimum.Lamethodeconsisteenungenerateurdepointsouslaformed'unedistributiondeprobabiliteselonlaquelledespointssontechantillonnes,quibiaisel'explorationenfaveurderegionsquiontdeforteschancesdecontenirl'optimum.L'algorithmegeneralestpresentesurlaFigure 6{1 .Supposonsquel'onsouhaite Figure6{1:Principedesalgorithmesaestimationdedistribution trouverlemaximumd'unefonctionFx,x=x1;:::;xn2Rn.Enl'absencedetouteinformationsurlalocalisationdumaximumoudesmaximax,ladistribu-tionderecherchepxestinitialiseecommeunedistributionuniformesurRn.Une

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137 populationdeindividusestcreeparechantillonnagedepx,etlavaleurdeFestcalculeepourchacundecespoints.Parmicespoints,bonspointssontchoisisenutilisantuneproceduredeselectionbaseesurlavaleurdeF.Alors,ladistributiondecespointsestevalueeetutiliseepourmettreajourladistributionderecherchepx.Laprocedured'echantillonnage/selection/miseajourestrepeteejusqu'acequ'uncritered'arr^etsoitsatisfait.L'estimationprecisedeladistributiondebonsindividusconstitueundesaspectscritiquesdesEDAs.Ils'agitdechoisirunmodeled'approximation,etd'utiliserlesdonneespourenestimerlesparametres.Laqualitedel'approximationdependdoncdedeuxfacteurs:lechoixdumodele,etlaprecisiondel'estimationdesparametres,lechoixdumodeleinuencantlaprecisiondel'estimation.Unmodeleexiblenom-breuxdegresdelibertepourraapproximeravecprecisionuneventailplusvastededistributionsmaisl'erreursurchacundesparametres,etdoncsurlespredictions,serapluselevee,pourunetailled'echantillonidentique.C'estlecompromisbiais/variance,toujourspresentlorsquel'ontraitedesproblemesd'approximation.Lechoixdumodeleaectel'ecacitedel'algorithmededeuxmanieres:unmodelepeuexiblerisquedeconduireadeserreursdemodelisationimportantes,doncalacreationetaucalculinutiledepointsdansdesregionsdefaibletness.Aucontraire,unmodeleexcessivementsophistiqueproduiradeserreursd'estimationpluseleveespourunetailled'echantillonidentique.Sil'onsouhaitereduirel'erreurd'estimation,ondoitaugmenterlatailled'echantillon,cequiapoureetd'accro^treleco^utdechaqueiterationd'EDA.Unbonmodelestatistiquedoitdonc^etrealafoissimplepeudeparametreset

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138 possederuneexibiliteadapteealatopologiedepx.Cesdeuxpreoccupationssontalabasedesapprochesproposeesdanscettedissertation:unpremieralgorithme,appeleUnivariateMarginalDistributionAlgorithm"UMDAutiliseunmodelesim-pledanslequellesvariablessontconsidereesindependanteslavaleurd'unevariablen'aectepaslafacondontlesautresvariablesagissentsurlatness.Cemodelepossedel'avantagedelasimplicite,maisn'estpasappropriepourrepresenterdesdistributionscomplexesdanslesquelleslesvariablesinteragissentfortementlesunesaveclesautres.Danscertainessituations,lacomplexiteapparented'unefonctionaapproximertientaunmauvaischoixdelabased'approximation.Unefaconderepresentercesfonctionsenapparencecomplexesparrapportalabased'approximationconsisteaeectuerunsimplechangementdebase.Ainsi,onpeutobtenirunemeilleureprecisionenconservantlem^emenombredeparametres.Parexemple,uneinnitedetermesestnecessairepourrepresenterunefonctionlineaireparuneseriedeFourier,alorsqu'unseultermesutsil'onutiliseunebasepolynomiale.Delam^ememaniere,ilestpossibled'ameliorerconsiderablementlaprecisiond'unmodelestatistiqueeninjectantdelaconnaissancesurlastructuredeladistributionpx.Danslecadredel'optimisationdestratiescomposites,commedansdenom-breuxautresdomaines,laconnaissancedelaphysiquedesproblemestraitespeut^etreutiliseeavecprotand'extrairedel'informationsurlaformedeladistributionpx.Eneet,dansdenombreusesapplications,lareponsedusystemedependdegrandeursmacroscopiquesquicapturentl'inuenceconjointedecertainesvariablessurlecom-portementglobal.End'autrestermes,cesgrandeurscaracterisentdescouplages

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139 physiquesdecesvariables.Ilsembledoncjustied'incorporerl'informationsurladistributiondecesgrandeursanderanerlemodelestatistique.Enparticulier,lesproprietesderigiditedesstratiesquientrentdanslecalculdelareponseglobaledelastructuresontcompletementdetermineesparunpetitnombredegrandeursmacro-scopiquesappeleesparametresdestratication",quisontdeniscommesommesdecosinusdesanglesdesbresdanslesdierentsplis.Lesvariablesd'orientationdesbresn'inuencentlafonctionco^utqu'atraverscesgrandeurs.Parconsequent,ladis-tributiondespointsselectionnespeut^etreexprimeesimplemental'aided'unnombrereduitdeparametresdansl'espacedesparametresdestratication.Surlabasedecesobservations,unnouvelalgorithme,appeleDouble-DistributionOptimizationAlgorithm"DDOAestpropose.Cetalgorithmcombineunmodelesimplemodeleunivariedeladistributiondesvariablesd'orientationetunmodeledeladistributiondesparametresdestratication,andeprendreencomptecertainesinteractionsentrelesvariables.6.3ContenudelatheseCettetheseestorganiseecommesuit: lechapitre 1 proposeuneintroductiongeneraleal'optimisationdestratiescomposites,eninsistantsurletypedeproblemesauxquelselleconduit.Lesmethodesexistantes,ainsiqueleursforcesetfaiblesses,sontpresentees.Lechapitretermineparunebreveintroductionsurlesmethodesd'optimisationevolutionnaires,etlesmethodesd'optimisationparalgorithmesaestimationdedistribution.

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140 Danslechapitre 2 ,lesalgorithmesaestimationdedistributionEDAsontpresentesdanslecontexteplusgeneraldel'optimisationstochastique.Lamiseaupointd'algorithmesestplaceedansuncadrestatistique:lesdierentscon-stituentsdesEDAsontdetailles,etlesenjeuxtheoriquesliesalaselectiondebonspointsetal'estimationdedistributionsontprecises.Lebutdecechapitreestd'introduirelesconceptsquivontpresideral'elaborationd'algorithmesdanslerestedecettethese. Danslechapitre 3 ,UMDA,unEDAbasesurunmodeleunivariedeladistribu-tiondespointsprometteurs,estmodiepourtraiterdesproblemesd'optimisationdestraties.L'inuencedediversparametresdel'algorithmeoriginaltailledepopulationetintensitedeselectionestobserveesurunproblemesimpledemaximisationdelarigiditelongitudinale.L'evolutiondelaperformanceavecl'augmentationdunombredevariablesestetudieedemaniereempiriqueettheorique.Onobservequel'ecacitedel'algorithmeestlimiteepardeserreursdeselectionlesvaleursoptimalespeuventnepas^etrepresentesdanslapopula-tiondespointsselectionnesetd'echantillonnagem^emesilemodelerepresenteparfaitementladistributiondespointsprometteurs,lesregionoptimalespeu-ventnepas^etrevisiteesdufaitdelatailleniedespopulations.Surlabasedecesconclusions,desameliorationsdel'algorithmesontpro-posees.Cesadditionsal'algorithmedebaseontpourobjectifcommundepreserverladiversitedeladistribution,etainsisacapacited'exploration.Troisapprochessonttestees:l'ajoutd'unoperateurdemutationsimilaireaceuxutilisesdanslesalgorithmesevolutionnairesconventionnels,l'adoptiond'une

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141 nouvelleproceduredemiseajourdesdistributionsincorporantunememoiredesgenerationsprecedentes,etuncontr^oleexplicitedeladegenerescencedesdistributionsonforcelesdistributionsmarginalesaconserveruneprobabilitenonnullepourchacunedesvaleursdelavariableconcernee.Enn,nousavonsappliqueUMDAatroisproblemesd'optimisationdestratiescomposites:lamaximisationsouscontraintesdelapremierefrequencepropred'uneplaquestratiee,laminimisationdelarigiditeencisaillementdanslecasoul'espacederechercheestdissymetrique,etlamaximisationdelaresistanced'unstratie,selonuncriterededeformationsmaximales.Cestroisproblemesontetechoisisavecsoindefaconarevelerlecomportementdel'algorithmesurdestopologiescaracteristiquesdesdiversproblemesrencontreslorsdelacon-ceptiondestructurescomposites:multimodalite,espacederechercheetroit,espacetrescontraint,etc.Surcesproblemes,l'algorithmeaetecompareadeuxautresalgorithmesstochastiques:unalgorithmegenetiquestandard,etunalgo-rithmederechercheparincrementslocauxstochastichill-climber,SHC.Pourcesproblemes,UMDAsemontreaussiecaceouplusecacequel'algorithmegenetique,etpossedeunavantagesurSHCquandleproblemeesttrescontraintoufortementmultimodal. Lechapitre 4 introduitunnouvelalgorithmeaevolutiondedistribution,ap-peleDouble-DistributionOptimizationAlgorithmDDOA,presentesurlaFig-ure 6{2 .L'ideeestd'ameliorerlaqualitedel'approximationdepxeninjec-tantdel'informationsurlastructuredeladistributiondespointsselectionnes.

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142 Figure6{2:AlgorithmeadeuxdistributionsDDOA L'algorithmeestbasesurl'etudededeuxdistributions:ladistributiondesvari-ablesd'orientationdesbres,representeeparunmodeleavariablesindependantes,etladistributiondesparametresdestratication,quicapturentdescouplagesphysiquesentrevariables,repesenteeparunesommedenoyauxgaussiens.Lesdeuxdistributionscooperentdelafaconsuivante:ungrandnombredepointsestgenereparechantillonnagedeladistributionpx1;:::;xn,puislapopu-lationnaleestobtenueenltrantcettepopulationatraversladistributionpV.Enfaisantvarierlerapportentrelapopulationdepointscandidatsetlataillenaledelapopulation,ilestpossibledereglerl'importancerelativedechacunedesdistributionsdanslacreationdenouveauxpoints.

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143 NousavonsappliqueDDOAatroisproblemesd'optimisationdestratiescom-posites:lamaximisationdelarigiditetransversaleA22sujetteaunecontraintesurlecoecientdePoisson,laminimisationducoecientdedilatationlongitu-dinalxtelquelapremierefrequenceproprenesoitpasinferieureaunecertainevaleurfmin,etlamaximisationdelaresistance,selonuncriterededeformationmaximale.Pourcestroisproblemes,DDOAsemontreplusecacequeUMDAetqu'unalgorithmegenetiquestandard,quinegerentpas,ougerentmallesdependancesentrevariables.Enoutre,l'avantagedeDDOAcro^taveclatailleduproblemenombredeplis,carl'eetdereductiondunombredevariablesparlepassagedansl'espacedesvariablesauxiliairesjouepleinement.L'inuencededeuxmecanismesdepreservationdeladiversitemutationetlimitationexplicitedesdistributionsmarginalesestensuiteexaminee.L'etuderevelequelamethodelaplusecacepouremp^echerlaconvergenceprematureedeladistributionconsisteaplacerdesbornesexplicitessurlesdistributionsmarginales.Undeseetsinattendusdel'algorithmeadeuxdistributionsestlaplusgrandestabilitedel'ecaciteparrapportaureglagedesparametresdevariabilitetauxdemutationoulimitesurlesdistributions.Lastrategieproposeeoreunmecanismedecorrectiondeladistributionderecherche:ellepermetdereduireouaugmenterlavariabiliteselonqu'elleestexcessiveouinsusante.Andedemontrerlecaracteregeneraldelamethodeproposee,uneversioncontinuedeDDOAestennpresentee,danslaquellealafoisladistributiondesvariablesduproblemeetladistributiondesvariablesauxiliairessontcontinues.

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144 Dansl'algorithmepropose,lesdeuxdistributionssontrepresenteesparunpro-duitdedistributionsnormalesunivariees.CetteversioncontinuedeDDOAesttesteesurunproblemegeneraldemaximisationd'unefonctionquadratique,su-jetteadescontraintesdenissantundomainefaisableetroitetcourbe.Commelescontraintesdenissentcontribuentpourunelargepartalatopologiedelatness,onchoisitunecontraintecommevariableauxiliaire.Lesresultatsdetestsmontrentquel'utilisationdecetteinformationameliorelaperformancedel'algorithmequandlapopulationinitialesetrouvedansledomainefaisable.Danslecascontraire,ellepeutfreinerlaconvergenceverscedomaine.Unestrategiepourtraitercessituationsestproposeeetmiseenuvre.Aveccettemodication,DDOAconvergeplusrapidementqueUMDAversl'optimum.6.4ConclusionsDanscettethese,desmethodesd'optimisationstatistiqueappeleesalgorithmesaestimationdedistributiononteteadapteesaucasparticulierdel'optimisationdestratiescomposites.Apresuneetudeapprofondieducomportementd'unalgo-rithmeexistantUMDA,unemethoded'optimisationpermettantd'incorporerdel'informationsurlastructureduproblemedanslebutd'ameliorerl'exactitudedumodelestatistiquedesregionsprometteusesestproposee.L'algorithmepropose,ap-peleDDOA,utilisedeuxdistributionspourguiderlarecherche:ladistributiondesvariablesdeconception,etladistributiondevariablesauxiliairesrepresentantl'actionconjointedeplusieursvariables.L'algorithmeamontresonecacitesurplusieursproblemesd'optimisationde

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145 composites,ouunereductiondutempsdeconvergenceaeteobserveeparrapportaUMDAetaunalgorithmegenetique.Lamethodeesttoutefoisgenerale,etpeut^etreappliqueeatoutproblemedanslequeldesvariablesauxiliairespertinentespeuvent^etreintroduites.

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152 S.-Y.Shin,D.-YCho,andB.-T.Zhang.Functionoptimizationwithlatentvariablemodels.InProceedingsoftheThirdInternationalSymposiumonAdaptiveSystemsISAS2001,pages145{152,2001. W.M.Spears,K.A.DeJong,T.Back,D.B.Fogel,andH.deGaris.Anoverviewofevolutionarycomputation.InPavelB.Brazdil,editor,ProceedingsoftheEuropeanConferenceonMachineLearningECML-93,volume667,pages442{459,Vienna,Austria,1993.SpringerVerlag,Berlin. D.ThierensandD.E.Goldberg.Mixingingeneticalgorithms.InS.Forrest,editor,ProceedingsoftheFifthInternationalConferenceonGeneticAlgorithms,pages38{45.MorganKaufmann,SanMateo,CA,1993. A.TodorokiandR.T.Haftka.Laminationparametersforecientgeneticoptimizationofthestackingsequenceofcompositepanels.InProc.7thAIAA/USAF/NASA/ISSMOMultidisciplinaryAnalysisandOptimizationSympo-sium,pages870{879,1998. A.A.TornandA.Zilinskas.GlobalOptimization.Springer-Verlag,Berlin,1987. S.W.TsaiandN.J.Pagano.Invariantpropertiesofcompositematerials.InS.W.Tsai,J.C.Halpin,andN.J.Pagano,editors,CompositeMaterialsWorkshop,1968. B.A.Turlach.Bandwidthselectioninkerneldensityestimation:Areview.Techni-calReport9317,C.O.R.E.andInstitutdeStatistique,UniversiteCatholiquedeLouvain,1993. S.VenkataramanandR.T.Haftka.Optimizationofcompositepanels{areview.InProceedingsoftheAmericanSocietyofComposites14thAnnualTechnicalConfer-ence,Fairborn,OH,pages479{488,1999. W.WieczorekandZ.J.Czech.Selectionschemesinevolutionaryalgorithms.InProc.oftheIntelligentInformationSystemsXI-IIS'02,Poland,pages185{194,2002. D.H.Wolpert.Ontheconnectionbetweenin-sampletestingandgeneralizationerror.ComplexSystems,6:47{94,1992. D.H.WolpertandW.G.Macready.Nofreelunchtheoremsforsearch.TechnicalReportSFI-TR-95-02-010,TheSantaFeInstitute,SantaFe,NM,1995. Z.B.Zabinsky.Stochasticmethodsforpracticalglobaloptimization.JournalofGlobalOptimization,13:433{444,1998.

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APPENDIXACOMPOSITELAMINATEOPTIMIZATIONA.1BasicmechanicsoflaminatedplatesA.1.1StinessmatricesLaminatedplatesaremadeupofanumbernplayers,alsocalledlaminaorplies,stackedontopofeachother.Usually,thematerialsusedforthelaminaaremadefrombersalignedaccordingtoaparticulardirectionunidirectionallaminaortwoperpendiculardirections,whichgivesthemorthotropicpropertiesofsymmetry.Sincethelayersarethin,oneusuallymakestheassumptionofplanestress,whichmeansthatalltransversestresses13,23,and33areneglected.Theconstitutiverelationforsuchmaterialsthenbecomes8>>>>>><>>>>>>:1122129>>>>>>=>>>>>>;=266666641 E1)]TJ/F23 7.97 Tf 10.494 4.813 Td[(21 E20)]TJ/F23 7.97 Tf 10.494 4.813 Td[(12 E11 E20001 G12377777758>>>>>><>>>>>>:1122129>>>>>>=>>>>>>;A.1Thestressesaregivenbytheinverserelation8>>>>>><>>>>>>:1122129>>>>>>=>>>>>>;=26666664E1 1)]TJ/F23 7.97 Tf 6.586 0 Td[(122112E2 1)]TJ/F23 7.97 Tf 6.587 0 Td[(1221012E2 1)]TJ/F23 7.97 Tf 6.586 0 Td[(1221E2 1)]TJ/F23 7.97 Tf 6.587 0 Td[(1221000G12377777758>>>>>><>>>>>>:1122129>>>>>>=>>>>>>;;A.2or=QA.3 153

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154 Whenthelaminaearerotatedwithrespecttoareferencecoordinatesystemx;y;z,thestinessmatrixinthatcoordinatesystem,Q,isobtainedbyQ=TTQTA.4whereTistherotationmatrixforstrainsfromanarbitrarycoordinatesystemx;y;ztothematerialdirections;2;312=Txy:T=26666664c2s2css2c2)]TJ/F22 11.955 Tf 9.299 0 Td[(cs)]TJ/F15 11.955 Tf 9.299 0 Td[(2cs2csc2)]TJ/F22 11.955 Tf 11.955 0 Td[(s237777775A.5withc=cosands=sin.Intheclassicallaminatedplatetheory,thebehaviorofalaminateisdescribedbythein-planeforceresultantNandthemomentresultantM:N=8>>>>>><>>>>>>:NxNyNxy9>>>>>>=>>>>>>;=Zh=2)]TJ/F23 7.97 Tf 6.586 0 Td[(h=28>>>>>><>>>>>>:xyxy9>>>>>>=>>>>>>;dzM=8>>>>>><>>>>>>:MxMyMxy9>>>>>>=>>>>>>;=Zh=2)]TJ/F23 7.97 Tf 6.587 0 Td[(h=28>>>>>><>>>>>>:xyxy9>>>>>>=>>>>>>;zdzA.6Afterintroducingthecinematicrelation=0+z;A.7where0isthemid-planeextensionalstrainandistheapproaimatedcurvature:=)]TJ/F28 11.955 Tf 11.291 46.028 Td[(8>>>>>><>>>>>>:@2w @x2@2w @y2@2w @x@y9>>>>>>=>>>>>>;;A.8

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155 oneobtainstherelationshipsbetweenthein-planeforceandmoments,andthemid-planestrainsandthecurvatures:N=A0+BA.9M=B0+DA.10whereA,B,andDaretheextensionalstinessmatrix,theextension-bendingcou-plingmatrix,andtheexuralstinessmatrix,respectively:A=Zh=2)]TJ/F23 7.97 Tf 6.587 0 Td[(h=2Qzdz=npXk=1tkQkA.11B=Zh=2)]TJ/F23 7.97 Tf 6.586 0 Td[(h=2Qzzdz=npXk=1tkQkA.12D=Zh=2)]TJ/F23 7.97 Tf 6.586 0 Td[(h=2Qzz2dz=npXk=1tkQkA.13A.1.2LaminationparametersInthiswork,wewillonlyconsiderlaminatesthathaveaplaneofsymmetryandthatsatisfythebalanceconditionforeach-ply,thereisa)]TJ/F22 11.955 Tf 9.299 0 Td[(-plyinthelaminate.Morespecically,wewillrestrictourstudytolaminatesoftheform[1=2=:::=n]s.ThesymmetryconditioncausesthecouplingstinessmatrixBtovanish.Thebalanceconditioncausesthein-planeextension-twistingcoecientsA16andA26tobezero,andD16andD26tobenegligeable.ThestinessmatricesofsuchlaminatescanbeexpressedintermsofhomogenizedgeometricalstinessquantitiesViandWi,

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156 calledlaminationparameters,andmaterialinvariantsU1toU5:8>>>>>>>>>><>>>>>>>>>>:A11A22A12A669>>>>>>>>>>=>>>>>>>>>>;=h266666666664U1U2U3U1)]TJ/F22 11.955 Tf 9.298 0 Td[(U2U3U50)]TJ/F22 11.955 Tf 9.299 0 Td[(U3U40)]TJ/F22 11.955 Tf 9.299 0 Td[(U33777777777758>>>>>><>>>>>>:1V1V39>>>>>>=>>>>>>;A.148>>>>>>>>>><>>>>>>>>>>:D11D22D12D669>>>>>>>>>>=>>>>>>>>>>;=h3 12266666666664U1U2U3U1)]TJ/F22 11.955 Tf 9.299 0 Td[(U2U3U50)]TJ/F22 11.955 Tf 9.298 0 Td[(U3U40)]TJ/F22 11.955 Tf 9.298 0 Td[(U33777777777758>>>>>><>>>>>>:1W1W39>>>>>>=>>>>>>;;A.15wherethelaminationparametersaregivenbyVf1;3g=2 hZh=20fcos2;cos4gdz=1 nnXk=1fcos2k;cos4kgA.16andWf1;3g=24 h3Zh=20fcos2k;cos4kgz2dz=1 n3nXk=1akfcos2k;cos4kg:A.17Theply-numberingconventionadoptedhereisshowninFigure A{1

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157 FigureA{1:Ply-numberingconventionforbalancedsymmetriclaminates ThematerialinvariantsaregivenbyU1=1 8Q11+3Q22+2Q12+4Q66A.18U2=1 2Q11)]TJ/F22 11.955 Tf 11.955 0 Td[(Q22A.19U3=1 8Q11+Q22)]TJ/F15 11.955 Tf 11.955 0 Td[(2Q12)]TJ/F15 11.955 Tf 11.955 0 Td[(4Q66A.20U4=1 8Q11+Q22+6Q12)]TJ/F15 11.955 Tf 11.955 0 Td[(4Q66A.21U5=1 8Q11+Q22)]TJ/F15 11.955 Tf 11.955 0 Td[(2Q12+4Q66:A.22Theequations A.16 and A.17 thatdenethelaminationparametersdeter-minefeasibleandinfeasibledomainsintheV-space:somecombinationsofV1,V3,W1,andW3arenotacceptable.Forinstance,ifweconsideranin-planeproblem,whereV1,V3onlyinuencetheobjectivefunction,V3=)]TJ/F15 11.955 Tf 9.298 0 Td[(1impliesthatthevalueofalltheplyanglesmustbe45,hencethevalueofV1cannotbechosenarbitrarily,buthastobe0.Theboundariesofthefeasibledomainhasbeenderivedforsimplecombinationsofthelaminationparameters:

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158 forV1;V3, MikiandSugiyama 1991 identiedthefeasiblespaceV32V12)]TJ/F15 11.955 Tf 11.955 0 Td[(1V31;A.23 similarly,forW1;W3,thefeasiblespaceisdenedbyW32W12)]TJ/F15 11.955 Tf 11.955 0 Td[(1W31:A.24Theserelationshipsdeneaparabolicfeasibledomainwhitearea,showninFigure A{2 FigureA{2:Feasibledomainforpairsoflaminationparameters Formorecomplexcombinations,onehastoresorttoapproximaterelationships. Diaconuetal. 2002b derivedapproximaterelationshipsforanarbitrarycombina-tionoflaminationparametersusingvariationalmethods.Inparticular,approximate

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159 boundariesforthecaseV1;V3;W1;W3aregivenby4W1)]TJ/F15 11.955 Tf 11.955 0 Td[(1V1)]TJ/F15 11.955 Tf 11.955 0 Td[(13A.254W3)]TJ/F15 11.955 Tf 11.955 0 Td[(1V3)]TJ/F15 11.955 Tf 11.955 0 Td[(13A.264W1+1V1+13A.274W3+1V3+13A.28V32V12)]TJ/F15 11.955 Tf 11.955 0 Td[(1A.29W32W12)]TJ/F15 11.955 Tf 11.955 0 Td[(1A.30V31A.31W31:A.32A.2MechanicalpropertiesusedinthisworkA.2.1Poisson'sratioInthecaseofisotropicmaterials,thePoisson'sratioisdenedastheratiobetweenthetransverseextensionoverthelongitudinalextensionforauniaxialloading22=12=0:=22 11:A.33Foranisotropicmaterials,suchascompositelaminates,thisquantitydependsontheorientation.Nevertheless,byanalogywithisotropicmaterials,aneectivePoisson'sratioehasbeendenedforlaminates:e=yy xxA.34e=A12 A22:A.35

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160 A.2.2CoecientofthermalexpansionTemperature-relatedexpansionsinorthotropicmaterialsaredierentinthelon-gitudinalandinthetransversedirections.Thistemperature-strainisexpressedbyT=M+FT;A.36whereTisthetotalobservedstrain,Mthemechanicalstraininducedbyappliedstresses,andFTisthestress-freethermalstrain.ForagivenstateofobservedstrainTandthermalstrainFT,theinducedstressesareobtainedfromthethematerialconstitutiverelation A.2 =Q)]TJ/F40 11.955 Tf 5.48 -9.684 Td[(T)]TJ/F40 11.955 Tf 11.955 0 Td[(FT;A.37expressedinthematerialdirections.Foraplyorientatedatanarbitraryanglekwithrespecttoareferencecoordinatesystem,thisequationbecomesxy=Q)]TJ/F40 11.955 Tf 5.48 -9.683 Td[(Txy)]TJ/F37 11.955 Tf 11.956 0 Td[(T)]TJ/F21 7.97 Tf 6.587 0 Td[(1FT12;A.38wherethethermalstrainsinthematerialdirectionsFT12aregivenby8>>>>>><>>>>>>:FT1FT2FT129>>>>>>=>>>>>>;=8>>>>>><>>>>>>:1209>>>>>>=>>>>>>;T:A.39

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161 ImposingthefreeloadconditionN=0,wegetN=Zh=2)]TJ/F23 7.97 Tf 6.587 0 Td[(h=2xydz=Zh=2)]TJ/F23 7.97 Tf 6.587 0 Td[(h=2Q)]TJ/F40 11.955 Tf 5.479 -9.684 Td[(Txy)]TJ/F37 11.955 Tf 11.955 0 Td[(T)]TJ/F21 7.97 Tf 6.587 0 Td[(1FT12=2npXk=1Q)]TJ/F40 11.955 Tf 5.48 -9.684 Td[(Txy)]TJ/F37 11.955 Tf 11.955 0 Td[(T)]TJ/F21 7.97 Tf 6.586 0 Td[(1FT12A.40=0:A.41HencenpXk=1QTxy=npXk=1QT)]TJ/F21 7.97 Tf 6.587 0 Td[(1FT12A.42and,introducingthecinematicrelations,AT0+BT0=npXk=1QT)]TJ/F21 7.97 Tf 6.586 0 Td[(1FT12=NT;A.43whereNTrepresentsthethermalloads.Theeectivecoecientsofthermalexpansionx=x=Tandy=y=TforasymmetriclaminateB=0cantheneasilybeobtainedas8>><>>:xy9>>=>>;=h 2[A])]TJ/F21 7.97 Tf 6.587 0 Td[(12664K1+K2V1K1)]TJ/F22 11.955 Tf 11.955 0 Td[(K2V13775;A.44wherethematerialconstantsK1,K2,andK3aredenedbyK1=U1+U41+2+U21)]TJ/F22 11.955 Tf 11.955 0 Td[(2K2=U21+2+U1+2U3)]TJ/F22 11.955 Tf 11.955 0 Td[(U41)]TJ/F22 11.955 Tf 11.955 0 Td[(2K3=U21+2+2U3+U51)]TJ/F22 11.955 Tf 11.955 0 Td[(2:A.45

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162 A.2.3FirstnaturalfrequencyConsiderasimply-supportedlaminatedplateasillustratedinFigure A{3 .In FigureA{3:Simply-supportedrectangularplate theclassicallaminationtheory,assumingsmalldisplacements,thenaturalvibrationfrequenciesaregivenbyfmn= 2p hr D11m L4+2D12+2D66m L2n W2+D22n W4;A.46whereLandWarethelengthandthewidthoftheplate,andmandndesignatethenumberofhalfwavesinthexandydirections.A.2.4StrengthLetusconsiderasingleplyunderastrainstatex,y,xy.Themaximumstraincriterionstatesthatfailureoccurswhenoneofthestrainsinthematerialdirectionsexceedtheultimatestrains.Thesafeconditionisexpressedby:1
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163 Thiscanbecondensedintheequations=minmaxt1 1;c1 )]TJ/F22 11.955 Tf 9.299 0 Td[(1;maxt2 2;c2 )]TJ/F22 11.955 Tf 9.298 0 Td[(2;s j12j>1A.48Therst-ply-failureFPFcriterionconsidersthatalaminnatehasfailedassoonasoneplyhasfailed.Thisissimplyexpressedby:mink=1;:::;nps>1A.49

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APPENDIXBBANDWIDTHSELECTIONFORDDOAB.1KerneldensityestimationSupposethatarandomvariablexhasaprobabilitydensitypx.ThetaskofndingthebestapproximationtopfromasetofNdatapointssampledfrompxiscalleddensityestimation.Twofamiliesofmethodsareavailabletoaddressdensityestimationproblems: parametricmethods supposedthatthedataoriginatefromaknownparametricfamilyofdensityfunctionsforexampleanormaldistributionwithmeanandstandarddeviation.Thegoalistondthevaluesoftheparametersthatbestdescribethedata; non-parametricmethods makenoatleastlessstringentassumptionsaboutthestructureofthedistributionbutletthedatadictateitsshape.Histogramsareanexampleofnon-parametricdensityestimates.InthekerneldensityestimateKDE,alsoknownasParzenwindowsestimate Dudaetal. 2000 ,theestimateddensityfunction^pisobtainedasthesumofsym-metricfunctions,orkernels,centeredaroundeachdatapointxi:px=1 NNXi=1Kx)]TJ/F37 11.955 Tf 11.955 0 Td[(xi:B.1 164

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165 ManydierentkernelsKucanbeused.ThemostcommonistheGaussiankernel:Ku=1 d=2dexp)]TJ/F37 11.955 Tf 10.494 8.087 Td[(uTu 2:B.2Theparameteriscalledthebandwidth.Itdeterminesthesmoothnessoftheestimate:ifissettoasmallvalue,theestimate^pwillexhibitspikesatthedatapoints;onthecontrary,ifalargevalueofischosen,localvariationsofpwillbesmoothedout,asshowninFigure B{1 ,wherethetruedistributionpisthesumof a=0:1 b=0:3 c=1:0FigureB{1:Eectofthebandwidth.Withsmallvaluesof,theestimatetendstoovertthedata,withlargevaluesof,localvariationsofparesmoothedout. threenormaldistributions.Thegureshowstheestimate^pbasedonasampleof100datapoints,forthreevaluesof.Clearly,abandwidthof0.3providesthebestapproximationofp.Whilethevalueofthebandwidthcandramaticallyaecttheaccuracyoftheestimate,determiningtheoptimalsettingisanon-trivialtaskwhentheunderlyingdistributionisunknown.Infact,whennopriorinformationaboutpisavailable,novalueofthebandwidthcanbedeemedbetterthanothers.Thenextsectionintroducestheheuristicusedinthiswork.

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166 B.2DeterminationofthebandwidthB.2.1EectofontheaccuracyApreliminarystudyoftheinuenceofthevalueofontheaccuracyofpVwasconductedinthecaseV=V1;V3.WeconsideredtheproblemofmaximizingA22ofabalancedsymmetriclaminate[1=2]s,withk2f0;5;10;:::;90g,subjecttoaconstraintonPoisson'sratio:maximizeA22suchthat0:48e0:52;B.3wheretheconstraintswereenforcedusingapenaltyapproachcf.Section 4{7 formoredetails.First,theexactdistributionofselectedpointspV1;V3wasemulatedby 1. computingthetnessofallpossiblecombinations[1=2]; 2. assigningaselectionprobabilityps1;2toeachpointofthedesignspace,proportionaltoitsrankbasedonthetness; 3. generatingapopulationofN=50;000pointsbysamplingfromps1;2andcomputingthevalueoftheauxiliaryvariablesV1andV3ofthecreatedpoints; 4. estimatingthedistributionpV1;V3oftheseselected"pointsbyakerneldensitymethod,usingasmallvalueofthekernel 1 .Thebestvalueofis 1Intheory,theapproximationerrorforacontinuousdensitypxconvergestozerowhenthenumberofsamplepointsNtendstoinnity Dudaetal. 2000 =0:1,andthebandwidthtendstozero.Inourproblem,thecardinalityofthespaceisnite,consequently,thebandwidthcannotbechosenarbitrarilysmall.

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167 determinedgraphically.Theideal"distributionpV1;V3obtainedbythisprocedureisshowninFigure B{2 FigureB{2:AccurateestimateofthedistributionpV1;V3ofselectedpointsfortheconstrainedmaximizationofA22. Theinuenceofontheaccuracyof^pV1;V3wastheninvestigatedbycalcu-latingtheaverageapproximationerrorforsevenvaluesofthebandwidth:=0.05,0.1,0.12,0.15,0.18,0.2,and0.25,andsamplesof=50selectedpoints.Theerrorwasestimatedbythedistancebetweentheapproximatedistribution^pV1;V3andthetruedistributionpV1;V3:d=1 mmXi=1j^pV1i;V3i)]TJ/F22 11.955 Tf 11.955 0 Td[(pV1i;V3ij;B.4wherethempointsV1i;V3iarearrangedaccordingtoa50by50gridin[)]TJ/F15 11.955 Tf 9.298 0 Td[(1;1]2.Toobtainanestimateoftheexpectedvalueoftheerror,theprocedureisrepeated50timeswithadierentsampleobtainedbylinearrankingselectionamongallpossiblepoints.Theevolutionoftheaverageerrordfor=50samplepointsisplottedinFigure B{3 .Theerrorisverylargeforsmallvaluesof,becausethemodelprovides

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168 FigureB{3:Inuenceofthebandwidthontheapproximationerrorforasamplesizeof=50points. poorestimatesofpV1;V3atpointsthatwerenotusedtobuildtheapproximation.Then,theerrorremainsstableinawiderangebetween0.12and0.2,beforeincreasingforlargevalues,becausethemodelthenfailstocapturelocalvariationsofthedensity.anexampleofapproximationobtainedwiththebestvalueofthebandwidth=0:18and=50isshowninFigure B{4 FigureB{4:Exampleofanapproximation^pV1;V3obtainedfor=50pointsand=0:18.

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169 B.2.2MaximumlikelihoodmethodOnethemostcommonparameterestimationmethodsusedinparametricdensityestimateisthemaximumlikelihoodmethod,whereoneseekstomaximizethelikeli-hoodofthedatabyadjustingthevalueofthemodelparameters.Thelikelihoodisdenedas:L=NYi=1^pxi;:B.5Thismethodcannotbedirectlyappliedtokerneldensityestimation,asthemethodwouldleadtothetrivialsolution!0Diracdeltaateachdatapoint.ToimprovethegeneralizationperformanceofKDE,across-validationmethoddescribedby Turlach 1993 wasimplemented.Themethodisamodiedmaximumlikelihoodapproach,wherethelikelihoodisreplacedbythepseudo-likelihood:~L=Yi=1p;iVi;B.6wherep;iisaleave-one-outestimationofthedensityatVi,basedonthe)]TJ/F15 11.955 Tf 10.943 0 Td[(1otherdatapoints:p;i=Xj=1j6=iKV)]TJ/F37 11.955 Tf 11.955 0 Td[(Vi:B.7B.3ApplicationinthecontextofDDOAThechoiceofthebandwidthisessentialforagoodestimationofpVandhencefortheeciencyofDDOA.Anupperbound 2 oftheoptimal"bandwidth 2Sincethepopulationconvergesastheoptimizationprogresses,thevariabilitywilldecrease,andsowill.

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170 canbefoundbyobtainingthevalueofthatprovidesthebestestimationofthedistributionofselectedpointsattherstgeneration.Thebandwidthwasallowedtotakevaluesfromf0:04,0.05,0.06,0.07,0.08,0.09,0.1,0.12,0.15,0.2,0.3,0.4g.Toreducethevarianceofthelikelihoodestimation,thetotallikelihoodLallover100independentrunswasusedasameasureofthegoodnessofthedensityestimation,foreachvalueof:Lall=100Yr=1Lr;whereLristhelikelihoodobtainedfortherthrun.Giventhesmallsamplesizeof=30andthesamplecreationprocedure=30pointsareselectedoutof=30pointsbylinearrankingselection,thesetofselectedpointsishighlylikelytocontainduplicatesofthesameindividual,hencecompromisingthevalidityofthecross-validation.Therefore,thecontributionofp;iViatpointsthathadbeenusedtoconstructtheapproximationwasignored.TheprocedurewasappliedforthersttwooptimizationproblemstreatedinChapter 4 :theconstrainedmaximizationofA22andtheconstrainedminimizationofthelongitudinalCTE,forn=6andn=12.TheloglikelihoodasafunctionofisshowninFigure B{5 fortheconstrainedmaximizationofA22problem,forthecasen=6theothercaseswoulddisplaythesametrend:itischaracterizedbyaverylowlikelihoodforsmallvaluesof<0:1,followedbyarelativelyatregion.Thisindicatesthatwhentoosmallkernelsareused,thepredictivepoweroftheapproximationiscompromised,whilethequalityoftheestimationisnotverysensitivetothechoiceofbeyond=0:2.Forthetwoproblems,thebestbandwidthwas=0:2forn=6and=0:15

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171 forn=12.Thistrendparallelstheevolutionoftheaverageerrorobservedinthe FigureB{5:LoglikelihoodLallasafunctionofforn=6,=30,constrainedmaximizationofA22 previoussection,thusconrmingthevalidityofthemethodforbandwidthselection.B.4Eectofthebandwidth:empiricalresultsTheinuenceofthechoiceofthebandwidthwasinvestigatedfortherstoptimizationproblemsaddressedinSection 4.4 :theconstrainedmaximizationofthetransversein-planestinessA22.Fourvaluesofthebandwidthweretried:=0:1,=0:15,=0:2,and=0:3.Whenintegratedintothefullalgorithm,thekerneldensityestimatecomponentinteractswithotherpartsofthealgorithm,consequentlythebestsettingofthebandwidthmaydierfromtheo-line"settingyieldedbythemaximumlikelihoodmethod.Itisthereforenecessarytotakeintoconsiderationthewaytheestimateisusedbythealgorithmtocreatenewpoints.Tothisaim,twotargetpointcreationschemesweretested,inassociationwiththedierentvaluesof:arejectionmethod,wherethefeasibilityoftargetpointswasenforcedbysamplingfromaparticularkerneluntilafeasiblepointwascreatedthedomainofallthepossiblevaluesofthe

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172 laminationparametersisdenedbytrigonometricrelationships,cf.Appendix A ,andamoreforgivingmethodthatallowedinfeasiblepointstobeusedastargets.Figure B{6 showsthemeanbesttnessforallthevariantstested,forn=6,andn=12.Surprisinglyenough,neitherthetargetpointcreationschemenorthebandwidthof an=6 bn=12 FigureB{6:Eectofthetargetpointcreationprocedureonthebesttness thekerneldensityestimatesignicantlyinuencedthealgorithm'sperformance:inthecasen=6,allbutthreeDDOAvariantsreject",=0:2and=0:3,andnoreject",=0:15outperformed,orhadcomparableperformancetoGAandUMDA;andinthecasen=12,allvariantsexhibitedaclearadvantageoverGAandUMDA.Onlyonevariant,reject"associatedwithabandwidthof0.3failedtoconsistentlyconvergetohightnessindividuals.Thisobservationconrmstheresultsofthemaximumlikelihoodbandwidthdetermination,whichshowedastableregionbetween=0:1and=0:2.Eventhoughthetwotargetpointcreationschemesdisplayedcomparableperfor-mance,itappearsthatthemeantnessofthenoreject"schemeismorestablewith

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173 respecttothechoiceof.Thiswasparticularlyvisibleforn=12:incontrasttothenoreject"schemeswhichexhibitednosignicantdierence,theperformanceofthereject"schemetendedtodeteriorateforlargervaluesof.Thiscanbeexplainedbythefactthatthereject"schemeredistributesthecontributionoftheinfeasibleportionofthekerneloverallthefeasibledomain,whereasthenoreject"schemeamountstoassigningallthatcontributiontotheboundary,assketchedinFigure B{7 .Consideringthatoptimumlaminatesoftenlieneartheboundaryofthefeasibledomain,andthatrejectsituationsmainlyoccurwhentheoptimumisclosetotheboundary,itseemssensibletofavorperipheralregionsovertherestofthedesignspace,hencethenoreject"approachseemstobeappropriate.Onthecontrary,theoverheadassociatedwiththereject"schemeisnotonlycostly,butdetrimentaltothealgorithm'sperformance. FigureB{7:Inuenceoftheboundariesonthesearchdistribution:reject"andnoreject"targetpointcreationschemeswhenx=5istheboundaryofthefeasibledomain.

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APPENDIXCLAMINATIONPARAMETERPROPORTIONALACCEPTANCE:EXPERIMENTALRESULTSTheinuenceoftheprocedureimplementedtosamplefromthetwodistribu-tionsusedinDDOA,theunivariatedistributionoftheprimaryvariablespx=px1px2:::pxn,andthedistributionofauxiliaryvariablespV1;V3wasinvesti-gatedforthefollowinglaminateoptimizationproblem:maximizeA22=hU1)]TJ/F22 11.955 Tf 11.955 0 Td[(U2V1+U3V3suchthatleuC.1inthecasen=6,withthematerialpropertiesgiveninTable 3{3 p. 68 .Theauxiliaryvariableprobabilityproportionalselectioncf.Section 4.2.2 iscomparedtothetargetpointapproachnorejectionscheme.Figure C{1 comparesthemeanbesttnessofDDOAwithprobabilityproportionalselectionforfourvalueofthepoolsizeandwiththetargetpointapproach=150tomeantnessofUMDAabandwidthof=0:2wasusedforthetwoDDOAvariants.ThebestperformanceisachievedbyDDOAwiththetargetpointapproach.Withtheprobabilityproportionalpointcreationmethod,DDOAdisplaysapoorperformance,regardlessofthevalueof:inthisDDOAimplementation,thelaminationparameterdistributionslowsdowntheoptimizationwithrespecttothebasicUMDAalgorithm. 174

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175 FigureC{1:LaminationparameterprobabilityproportionalacceptancecausesDDOAtobeveryconservative,thusresultinginslowerconvergencetotheoptimumthanUMDA.

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APPENDIXDMULTIOBJECTIVEOPTIMIZATIONOFLAMINATESTheattractivequalitiesofcompositematerialsaretheirveryhighstiness-to-weightorstrength-to-weightratios.However,thebulkandprocessingcostofthesematerialsincreasesdramaticallywithperformance.Althoughinaerospaceapplica-tions,designisdrivenbystructuralweightconsiderations,costbecomesafactorinapplicationsintendedforthegeneralpublic,suchasautomobiles.Insuchcases,itmaybeadvantageoustoresorttocombinationsofecientbutexpensive,andlessex-pensivebutlessstimaterialsinordertoreducecostwhileensuringhighperformance.Combiningthetwomaterialsmakescostsavingspossiblewithoutcompromisingper-formancesignicantly.Findingthebestcombinationsrequiresthetrade-obetweenperformanceandcosttobecarefullystudied.Thiscanbedoneintheframeworkofmultiobjectiveoptimization.Thepresentchapterinvestigatesthecombinationoftwomaterialsforbalancingcostandweight.Themethodologyisdemonstratedonasimplelaminateoptimizationproblem.D.1ProblemdescriptionAsimplysupportedlaminatedplateoflengtha=36inandwidthb=30inistobeoptimizedforminimumweight,W,andcost,C,subjecttoalowerboundfminof25Hzontherstnaturalfrequencyf.Acompromisesolutionbe-tweenthosetwocompetingobjectivescanbeachievedbycombiningtwomateri-als:graphite-epoxyandglass-epoxy.ThematerialpropertiesaregiveninTable 176

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177 D{1 source http://composite.about.com .Thestiness-to-weightratioofgraphite-epoxyisaboutfourtimeshigherthanthatofglass-epoxy,withE1==345againstE1==87:5.Howeveritisalsomoreexpensive,withacostperpoundthatisabout8timeshigherthanthatofglass-epoxy.Iftherstpriorityisweight,thengraphite-epoxywillbepreferred;whileifcostisparamounttheoptimumlaminatewillobviouslycontainglass-epoxyplies.Thedesignofthissimplerectangularplateleadsustostudythetrade-obetweenthetwoobjectivefunctions:weightandcost.Theplyorientationcantakeonasetof19valuesrangingfrom0to90degreesinstepsof5degrees.Hence,theproblemconsistsinndingthenumberofpliesandthestackingsequenceofthelaminate,whichistheorientationandmaterialofeachply.Thelaminateissymmetricandbalancedforeveryangledierentfrom0and90,theangle)]TJ/F22 11.955 Tf 9.298 0 Td[(isalsopresentinthelaminate.Theseconstraintsareimposedtoensurethemanufacturabilityofthestructureandareusualinlaminatedesign.Theysimplifytheanalysisconsiderably,asthesymmetryassumptionimpliesthattheextensional-exuralcouplingmatrixBoftheclassicallaminationtheoryiszeroandthebalanceconstraintsuppressesthenormal-shearextensionalcouplingA16=0andA26=0andminimizesthemagnitudeofthetorsion-bendingcouplingtermsD16andD26.Themultiobjectiveoptimizationproblemcanbeformulatedasfollows:Minimize:WandCbychanging:theorientationiandthematerialmiofthepliessuchthat:g=f fmin)]TJ/F15 11.955 Tf 11.955 0 Td[(10D.1

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178 TableD{1:Materialpropertiesofgraphite-epoxyandglass-epoxy Graphite-epoxyGlass-epoxy LongitudinalmodulusMsi,E120.016.3TransversemodulusMsi,E21.301.29In-planeshearmodulusMsi,G121.030.66Poissonmodulus,120.30.27Densitylb/in3,0.0580.072Thicknessin,t0.0050.005Costfactorlb-1,c8.01.0 Usingclassicallaminationtheory,therstnaturalvibrationfrequencyisgivenbythefollowingexpression:f= 2p hr D11 a4+2D12+2D66 a2b2+D22 b4D.2whereDistheexuralstinessmatrix,istheaveragedensityandhthetotalthicknessoftheplate.D.2IntroductiontomultiobjectiveoptimizationInmultiobjectiveoptimization,mobjectivefunctionsF1xtoFmxarecon-sideredsimultaneously.Ingeneral,itisnotpossibletondapointxthatminimizesallobjectivesbecausesomeobjectivesconictwithoneanother.Instead,weseekpointswhereonecannotimproveoneobjectivewithoutcausinganotherobjectivetodeteriorate.Suchpointsarecallednon-dominatedbecauseonecannotndpointsthatoutperformtheminallobjectivefunctions.Non-dominatedpointsaresaidtobePareto-optimal.Thatis,apointxisPareto-optimalifandonlyifthereisnopointxwiththecharacteristics:FixFixforalli,i=1;2::::;m

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179 andFix
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180 method,consistsinrepeatingthesingle-objectiveoptimizationofthecompositefunc-tionjustdescribedusingdierentacombinationoftheweightingfactorseachtime,thusobtainingadierentPareto-optimalpointeverytime.Theconstraintapproachkeepsoneobjectivefunctionandreformulatesallotherobjectivesasconstraints.BychangingtheweightingcoecientsofFortheconstraintlimits,thewholeParetosetcanbefound.Alternatively,thereareevolutionaryalgorithmsthatapproximatetheParetosetinasingleoptimization.Forexample, HajelaandLin 1992 investigatedsuchgeneticstrategiesinthecontextofstructuraloptimization.Inlaminateoptimization,severalapproacheshavebeentakentoapproximatetheParetosurface. Adalietal. 1996 usedaweightingmethodtosolvethethree-objectiveproblemofmaximizingprebuckling,bucklingandpostbucklingstrengthofalaminatedplate. KumarandTauchert 1992 observedthatnon-convexParetofrontcanexistinthecaseofmaximumin-planeorbendingstrengthorstinessmaximization,andthereforerecommendedusingtheweightingmin-maxmethod,whichisamodiedmin-maxalgorithmwherethecontributionofeachobjectivetothedistancebetweenaparticulardesignandtheidealpointisgivenadierentweight.InthisworktheParetosetisgeneratedbyoptimizingaconvexcombinationofthetwoobjectives,weightWandcostCintoacompositefunctionF:F=C+)]TJ/F22 11.955 Tf 11.955 0 Td[(WD.3for01.Thetrade-ocurvebetweenweightandcostisthenconstructedbyperforming

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181 asuccessionofthefollowingsingle-objectiveoptimizationsforseveralvaluesof:Minimize:Fbychanging:theorientationiandthematerialmiofthepliessuchthat:g=f fmin0D.4Awell-knownlimitationofthemethodisthatitfailstocaptureconcaveportionsoftheParetoset.Duecarewillbetakenwheninterpretingtheresults.D.3GeneticalgorithmGeneticalgorithmsGAareinspiredbyDarwin'sprincipleofevolution,whichstatesthatapopulationofindividualsiscapableofadaptingtoitsenvironmentbe-causeindividualsthatpossesstraitsthatmakethemlessvulnerablethanothersaremorelikelytohavedescendentsandthereforetopassontheirdesirabletraitstothenextgeneration.Onecanthinkofthisprocessofadaptationasanoptimizationprocessthatprobabilisticallycreatestterindividualsthroughselectionandrecom-binationofgoodcharacters.Geneticalgorithmsaresimpliedcomputermodelsofevolution,wheretheenvironmentisemulatedbytheobjectivefunctiontomaximize,andthestructuretooptimizeplaystheroleoftheindividuals.AowchartofageneticalgorithmispresentedinFigure D{1 .GAsstartbyinitializingapopulationofindividualsatrandom.Eachindividualencodeofapar-ticularcandidatestructureintheformofoneorseveralchromosomes,whicharestringsofnitelength.Thentheobjectivefunctionofeachindividual,calledtnessfunctioninthecontextofevolutionarycomputation,iscomputed.Thetnessofeach

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182 individualdeterminesitsprobabilityofbeingselectedforreproduction.Recombina-tionandmutationoperatorsarethenappliedtotheselectedindividualstheparentstocreateapopulationofchildren.Finallyasurvivalruledeterminestheindividualsamongtheparentandchildpopulationthatwillbekepttoformthenewpopulation. FigureD{1:Geneticalgorithm D.3.1LaminateencodingIntheproblemunderconsideration,thedesignvariablesarethenumberofplies,thematerialandtheberorientationofeachply.Theplatedimensionsarexed.Hencealaminatecanberepresentedbytwostringschromosomes,onefortheberorientation,theotherforthematerialwhereeachcharactergenedescribestheorientationormaterialofaparticularply.Variablethicknessisachievedby

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183 allowingemptyplies.Sinceonlybalancedsymmetriclaminatesareacceptable,thenumberofgenesrequiredtorepresentalaminateisonlyaboutonequarterofthenumberofplies,becauseonlyhalfofthelaminateneedstobeencodedandbalanceisenforcedbyusingpairsofplieswithoppositeangles.Sincethebalanceconstraintdoesnotaect0and90plieswhenloadingswithrespecttothelaminatecoordinatesareconsidered,onlytheseanglesareallowedtoappearasindividualplies,therebysuppressingunnecessarylayersandprovidingmoreexibilityinthedesign.The19possibleanglesfrom0to90in5steps,aresimplycodedasintegersbetween1and19,with1correspondingto0and19to90.EmptypliesweredenotedbyE.Similarly,theplymaterialwascodedbytwointegers:1forgraphite-epoxyand2forglass-epoxy.Asanexample,theindividual:Orientation[4/7/12/1/3/E/E]Material[1/2/2/1/1/E/E]willbedecodedas:[15=30=55=0=10]s,wherenormalcharactersdenotegraphite-epoxyandboldcharactersdenoteglass-epoxy.D.3.2FitnessfunctionGeneticalgorithmsdiscriminatebetweenpromisingandpoorindividualsbycom-paringallmembersofapopulationonthebasisofascalarscore,calledtnessfunc-tion,whichsummarizestheindividualperformances.Geneticalgorithmsdonotnat-urallyaccommodateconstraints,howeverseveralheuristicscanbeusedtodirectthesearchtowardfeasibleregions.Themosteectiveandinexpensivewayofenforc-ingconstraintsistousedatastructuring,i.e.tohard-codetheconstraintintothe

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184 chromosomes.Forexample,inthiswork,symmetryandbalanceconstraintsareau-tomaticallymetbecausethedecodingproducesonlysymmetricbalancedlaminates.However,mostconstraintscannotbeeasilyhard-codedintothechromosomes.Forsuchconstraints,twomaintechniques:repairandpenaltyapproachesareavailablesee Michalewiczetal. 1996 .Repairstrategiesuseheuristicstoreplaceanindi-vidualthatviolatesconstraintswithitsprojection"ontotheconstraints,sothatitsgeneticcontentisnotlostaltogether.Repairstrategieshaveproveneectiveonmanyproblems.Forinstance, TodorokiandHaftka 1998 usedaphenotyperepairstrategytoenforcebalanceandcontiguityconstraints.However,thereisnostandardrepairmethod,andrepairinganindividualcanbeverydicultwhentheconstraintisacomplexfunctionofthedesignvariables.Inourcase,thefrequencyconstraintfallsintothatcategory,consequentlyrepairisnotpractical.Weinsteaduseapenaltyapproach,wherethetnessofindividualsthatviolateconstraintsisdecreasedtoac-countforthecostofsatisfyingtheconstraint.Inthiswork,weadoptedthefollowingformulationforthetnessfunctionF:F=)]TJ/F22 11.955 Tf 9.299 0 Td[(F)]TJ/F22 11.955 Tf 11.956 0 Td[(gifg0feasible)]TJ/F15 11.955 Tf 9.299 0 Td[(1:1F+pgotherwiseinfeasibleD.5wherepdenotesthepenaltyparameter,whichisselectedthroughexperimentstobehighenoughtoensurethatthedesignwiththelowestvalueofFdoesnotviolatetheconstraint.Themultipliertakentobe0.01intheexpressionforfeasibledesignisusedtorewarddesignsthatsatisfytheconstraintwithlargermargins.

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185 D.3.3GeneticoperatorsThreetypesofoperatorsareusedtoproducenewindividualsfromaparentpopulation:selection,recombinationandmutation.Theroleofselectionistoidentifyindividualsthatpotentiallycontainpartialsolutionsbuildingblocks,recombinationmixesgeneticmaterialfromtheparentstogeneratechildrenwiththehopethatgoodtraitsfromallparentsgetcombined.However,ifonlythesetwooperatorsareused,thepopulationlosesgeneticdiversity,andoncethepopulationhasbecomeuniform,nomoreprogressispossible.Topreventprematureconvergence,aperturbationoperatorcalledmutationisusedtoaddarandomexplorationcomponenttothealgorithm.Selection.Oneofthekeyfactorsofevolutionaryalgorithmsisselection.Inthiswork,rankproportionalselectionwasused.Themainadvantageofrankbasedselectionisthatitisinsensitivetotheobjectivefunctionscaling.Itallowstheselec-tionpressuretoremainconstantthroughouttheoptimization,incontrasttotnessproportionalselection,whichlosesitsdiscriminatingcapacitywhenallindividualsbecomeclosetotheoptimum.Crossover.Theroleofrecombination,orcrossover,istoexchangegeneticma-terialbetweentwoparentstocreatetwochildrenwhocombinetraitsfrombothpar-ents.Themostcommonformsofcrossoveroperatorsareuniformcrossover,one-pointandtwo-pointcrossover.Theydierintheamountofgenemixingtheyaccomplish.Inuniformcrossover,eachgeneoftherstchildischosenwithequalprobabilityfromeitherparent,thesecondchildreceivesthegenefromtheotherparent.Inone-point

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186 crossover,abreakpointischosenrandomlyinthechromosomes,andtheospringsarecreatedbyswappingtheparents'substrings.Two-pointcrossoverchoosestwobreakpointsatrandom,andswapsthesubstringlimitedbythesetwopoints,asdepictedinFigure D{2 Parent1:4_^71213_^210Parent2:4_^16935_^73givesChild1:47121373Child2:416935210 FigureD{2:Two-pointcrossover Inthiswork,thetwo-pointcrossoverwasusedbecauseunlikeone-pointanduni-formcrossovers,itdoesnotbreaklinkagebetweeninnerandouterplies.Itrespects"cooperationbetweentheseplies.Itshouldbenotedthatoutermostpliesareprepon-derantinthebendingbehavioroftheplate.Mixingoccursmainlywithinmid-rangeplies.Inthecaseofsimultaneousoptimizationoftheplyanglesandmaterials,therearetwooptionsforapplyingcrossover:thetwochromosomescanbetreatedseparatelydistinctbreakpointsortogether.Extensivetestingconductedpriortothisstudyshowedthattheformeroptionwasmoreecient.Mutations.Traditionally,mutationoperatorsareviewedasbackgroundop-eratorinthegeneticalgorithmcommunity.However,mutation,inabroadsenseisoftenaninvaluabletoolforexploringnewregionsofthedesignspaceandpreventprematureconvergence.Whenmutationoperatorsarespecicallytailoredtoaclassofproblems,considerabletimesavingscanbeachieved.Threetypesofmutationwereusedinthiswork.

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187 SimplemutationAstandardmutationoperatorisappliedtobothchromosomesofeverychildindividualwithsomeuser-speciedprobabilitiespomorientationandpmmmaterial.Mutationchoosesageneatrandomandswitchesitsvaluetoanyotherpossiblevalue,asillustratedinFigure D{3 Beforemutation:471213210Aftermutation:47313210 FigureD{3:Simplemutation PermutationThepermutationoperatorwasintroducedby LeRicheandHaftka 1993 tosolvestackingsequenceoptimizationproblems.Itisespeciallyusefulbe-causeitchangesthebendingpropertiesofalaminatewithoutaectingitsin-planeproperties.Thepermutationoperator,appliedwithprobabilitypptoachromosome,choosestwogenesatrandominthatchromosomeandipstheorderofthesubstringcontainedinsiderthesetwogenesseeFigure D{4 Beforepermutation:4_^7121_^3210Afterpermutation:4_^1127_^3210 FigureD{4:Permutation AdditionanddeletionVariablenumberofplieswasaccomplishedbyallowingthegenestotakeaspecialvalueE,whichdenotesanemptyply.Thatvaluecanonlybesetorliftedbydedicatedmutationoperatorscalledadditionanddeletion,whichareappliedwithuser-denedprobabilitiespaandpd.AdditionanddeletionoperatorsareillustratedinFigure D{5

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188 AdditionBeforeaddition:471213EEAfteraddition:47121713EDeletionBeforedeletion:4712137EAfterdeletion:47137EE FigureD{5:Additionanddeletion D.3.4ImplementationAgeneticalgorithmformultiobjectiveoptimizationwasdevelopedusingtheGAtemplateforlaminateoptimizationdevelopedby McMahonetal. 1998 .Thatpackageprovidesaframeworkforbuildingalgorithmsthatcantackleawidevarietyofcompositestructuredesignsituations,includingmultiplematerials,multiplelami-nates,continuousanddiscretevariablesforstackingsequenceandgeometricvariables.ItiscomprisedofaGAmoduleandapackageofGAoperators.TheGAmoduledenesadatastructurethatfacilitatesmanipulationofpopulationsorindividualsinanobject-orientedapproach.Theprogramwasmodiedtoincorporatethetwo-pointcrossoverappliedindependentlytotheorientationandmaterialchromosomes.Thedecodingprocedurewasmodiedtoimplementthebalanceconstraintasdescribedpreviouslypairsofanglesofoppositesignsareusedexceptfor0and90.D.4ResultsFirsttheminimumoftheweightandcostobjectivefunctionswerefoundusingthegeneticalgorithm.Theoptimizationstartedwithapopulationof10designsmadeof44plieselevengeneswhoseorientationandmaterialweresetrandomly.TheGAwasappliedwiththeparametersshowninTable D{2 allprobabilitiesareprobabilitiesofapplicationperchromosome.Thesevaluesweredeterminedbytrialanderrorinordertomaximizereliability,whichistheprobabilityofreachingthe

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189 optimumforagivennumberoffunctionevaluations.Theyareusedinallcasesinthisstudy. TableD{2:Parametersassociatedwiththegeneticoperators CrossoverprobabilityMutationproba-bility,orientationMutationproba-bility,materialsPermutationprobabilityAdditionprobabilityDeletionprobability 1.00.30.20.20.050.1 Theminimumweight=1:0wasfoundtobe6.89lb.Itisreachedwhenallpliesaremadeofgraphite-epoxy.Theoptimumdesignisthen[505=0]sandhasarelativecostof55.12.Theminimumcost=0:0was16.33,70%lessexpensivethantheoptimumoftheweightobjectivefunction.Theoptimumdesignis[5010=0]scomprisedexclusivelyofglass-epoxy.Itsweightis16.33lb,whichrepresentsanincreaseof137%overtheoptimumoftheweightobjectivefunction.InordertoconstructtheParetofront,theweightingfactorawasvariedfrom0.0to1.0andthecompositeobjectivefunctionFwasminimizedusingtheGA.TheoptimumdesignsobtainedaswellastheirweightandcostaresummarizedinTable D{3 .Theoptimumdesignsaremainlymadeof50pliesinordertomaximizetherstvibrationfrequencyoftheplate.Insomecasesa0or90plyispresentinthecorelayersofthelaminate.Althoughthesepliesdonotcontributemuchtothefrequency,itisadvantageoustousethembecauseunlikeotheranglestheydonothavetocomeinpairstosatisfybalance,therebysavingunnecessaryadditionalweightandcost.Whenasinglematerialisused,the0and90pliesalwaysappearintheinnerlayers,wheretheyaretheleastdamagingfortheperformanceoftheplate.Whentwomaterialsareused,theyareplacedintheinnermostlayerofeachmaterialblock.

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190 Forthesamereason,thelessstiglass-epoxylayers,whentheyhavetobeused,alwaysappearintheinnerlayers.ThiscreatesthesandwichtypecompositeshowninFigure D{6 wherethestructuralfunctionisassuredbythestigraphitelayers,placedontheoutside,wheretheircontributiontotheexuralpropertiesofthelaminateismaximal,whileinnerlayersaremerelyusedtoincreasethedistanceoftheouterpliesfromtheneutralplane. TableD{3:Optimumdesignsoftheminimizationofthecompositeobjectivefunction WeightingfactorCostWeightFirstnaturalfrequencyfmin=25HzStackingsequenceplainnumbers:graphite,boldfacednumbers:glass 10.0016.3316.3325.82[5010=0]s 20.7020.9012.1425.10[50=507]s 30.8027.8210.2825.88[502=505]s 40.8731.289.3525.25[502=90=504]s 50.9038.968.2725.38[503=90=502=0]s 60.9643.208.1226.07[504=502]s 71.0055.126.8925.14[505=0]s 50,glass-epoxy planeofsymmetry 50,graphite-epoxy FigureD{6:Sandwichstructureoftheoptimumdesign2Table D{3 .Thestructuralfunctionismainlyassuredbythegraphite-epoxylayersplacedontheoutside,whiletheroleoftheinnerglass-epoxylayersistoincreasethedistanceoftheouterpliestotheneutralplane. Design5isparticularlyinstructivebecauseitrevealshowtheGAwasabletoachievecostsavingswhilepreservingperformance.Itexhibitsthesandwichtypestructurecharacterizedbystiergraphite-epoxypliesontheouterlayers,butinside

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191 eachlayerweightandstinessaretradedbyusingthe50stackswheretheyarethemosteectiveandpaddingwithindividual0and90pliesthatgivemoreexibilityinthedesign.Figure D{7 showstheParetofrontobtainedusingtheweighedsummethod.Thesolidblacklinerepresentsthesetofsolutionsofthecompositeobjectivefunctionforthedierentvaluesof.Thevarioussymbolsshowallthefeasibledesignsthatweregeneratedduringthesearch.TheParetofrontthesetofallthenon-dominatedsolu-tionscorrespondstothelowerleftenvelopeofallthedesignpointsintheweight/costplane.Outofthesampledfeasiblepoints,thesevendesignsmakinguptheParetofronthavebeenfoundbytheweightedsumapproach.ApeculiarityofthecurrentParetocurveisitsconcavityatPoint6,Intheory,theweightingmethodusedtogenerateitcannotndsuchpoints,asillustratedinFigure D{8 .ThedashedlinesindicatethecontoursofthecompositefunctionFintheC;W-plane,withvaluesdecreasingasonegetsclosertotheorigin.AconceptualconcaveParetofrontisrepresentedbythesolidline.Whenthefactorisvariedfrom0to1theorientationofthedashedlinesgoesfromverticaltohorizontal,theconcavepartofthecurveneveryieldstheminimumofF.OnlypointsontheconvexhulloftheParetosetcanbedetermined.Points1,2,4and5canbefoundbythismethod,butnotpoint6.However,for=0:90,thealgorithmalternativelyyieldeddesign5ordesign6.Infact,outofatotalof50runs,itconverged29timestodesign5and21timestodesign6.Thetrueoptimumisdesign5,whichgivesF=11:34,whereasthedesign6evaluatesatF=11:63,butdesign5issubstantiallymorediculttondbecauseit

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192 FigureD{7:Paretocurvefortheminimumweightandcostdesignofthelaminatedplate.Thesolidblackcurvejoinsnon-dominatedpoints. islocatedinadeepvalley:allofitsneighborsareeithersubstantiallyheavierbecausethe0and90pliesarenotpresent,orviolatethefrequencyconstraint.Incontrast,design6isextremelyrobustbecausethesingle-plystacksarenotusedandthemarginofsafetyforthefrequencyislarge.Inthiscase,convergencetoalocaloptimumisdesirable,asitrevealsconcavepartsoftheParetoset.DuringtheapproximationoftheParetoset,wefoundthatitwascriticaltoadjustthepenaltyparameterpinordertondallthenon-dominateddesigns.Whenpwastoohigh,thesearchwasbiasedtowardnonoptimalthickerlaminates,whereastoo

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193 C W decreasingF=W+)]TJ/F22 14.346 Tf 14.346 0 Td[(C 1 2 3 4 5 FigureD{8:TheweightingmethodisnotappropriatetocaptureconcaveParetofrontingeneral.Points1,2,4and5canbefoundbythismethod,butnotpoint3. lowvaluesofpcausethelaminatetocollapsetozerothicknesslaminatesbecausethestructuralrequirementwasdominatedbytheweight/costconsiderations.Sincethepenaltymustbescaledtothemagnitudeoftheobjectivefunction,whichdependedon,agoodvalueofphadtobedeterminedbytrialanderrorforeachofitsvalues.Wefoundthatagoodvalueofpwas15.0for=0:0and8.0for=1:0.Theformulationoftheweightingmethodcanbeperfectedbynormalizingthetwoobjectives,sothatthemagnitudeofthecompositefunctiondoesnotchange.Thiscanbeaccomplishedbydividingeachobjectivebyitsmaximumvalue:^F=^W Wmax+)]TJ/F15 11.955 Tf 13.115 0 Td[(^C CmaxTheParetotrade-ocurvecanbeusedtohelpthedesignerdeterminetheoptimalcongurationforhisproblem.Thenalchoiceofthebestdesignwilldependonadditionalinformationthatwillenablehimtoassignprioritiestothetwoobjectives.

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194 Thereisnosinglebestdesign:dependingontheapplicationthatisconsidered,thechoicewillbedierent.Forexample,ifweightandcostareofsimilarimportance,thedesign5obtainedwith=0:93inTable3maybeattractive.Theweightis20%abovetheminimumweight,butthecostissignicantlyreducedby30%.D.5ConcludingremarksThemethodusedinthischapterforsolvingthemulti-objectiveoptimizationproblemthatderivesfrommulti-materialcompositedesignissimple.Itisbasedonastandardgeneticalgorithmforcompositelaminateoptimization.ThePareto-optimalfrontwasconstructedbyoptimizingaseriesofcompositeobjectivefunctionlinearlycombiningweightandcost.ThemethodwasshowntobeabletocaptureimportantfeatureoftheParetotrade-ocurveforthattwo-objectiveproblem,providingthedesignerwithahelpfuldecision-makingtool.Anexampleofaplatedesignedtomin-imizeweightandcostsubjecttoaconstraintontherstnaturalfrequencyillustratedtheapproach.Whencostwasaprimaryconsiderationtheplatewasmadefromglass-epoxy,andwhenweightwasaprimaryconsideration,itwasmadeofgraphite-epoxy.CompromisedesignsareeasilyselectedfromtheParetotrade-ocurve.Inparticular,thestudyrevealedthatsubstantialcostsavingscanbeachievedwithonlyamoder-ateweightincreasebyreplacingsomeoftheexpensivegraphite-epoxyinnerlayersbyglass-epoxy.Additionalworkremainstobedoneinordertoimprovetheeciencyoftheoptimization.Performancemaybemarginallyincreasedbychangingtheweightingmethodandpenaltyfunctionformulationsinordertoimprovetherobustnessof

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195 theschemeagainstchangesintheweightingfactor.Preliminaryinvestigationindicatesthatnormalizingtheobjectivefunctionscansignicantlyreducetheamountofpenaltyparametertuningneededanddistributethenon-dominatedpointsmoreuniformlyintermsoftheparameter.Theweightingmethodappliedinthisworkrequiresmultiplenumberofrunsandcanbecomputationallyexpensive.Thereexistmultiobjectiveevolutionaryalgo-rithmsMOEAapproximatetheParetosetinonerunbyusingParetodominanceintheselectionandadiversitypreservingschemeintheobjectivespace.Anotherwayofreducingthenumberofanalysesrequiredistoimprovetheeciencyofeachindivid-ualsingleobjectiveoptimization.Ourpurposeistodevelopagenericalgorithmthatcanbeusedformultiobjectiveoptimization,butforsingleobjectiveoptimizationaswell.Themainchaptersofthisdissertationareconcernedwiththeinvestigationofecientsingleobjectivelaminateoptimizationmethods.

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BIOGRAPHICALSKETCHLaurentGrossetwasborninCombourg,France,onMay3,1977.HereceivedaBachelorofEngineeringdegreeinmechanicalengineeringfromtheInstituteofAppliedSciencesofRouen,France,2000.Duringhisengineeringstudies,hecarriedoutseveraltrainingperiodsinindustry,inparticularathree-monthinternshipinthemechanicaltestingdepartmentofDaimlerChryslerinStuttgart,Germany,andasix-monthinternshipinthesteamturbinedesigndepartmentatHitachiWorks,inJapan. 196


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Title: Optimization of Composite Structures by Estimation of Distribution Algorithms
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OPTIMIZATION OF COMPOSITE STRUCTURES
BY ESTIMATION OF DISTRIBUTION ALGORITHMS














By

LAURENT GROSSET


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2004

































Copyright 2004

by

Laurent Grosset

















I dedicate this work to my parents.















ACKNOWLEDGMENTS

I want to express my gratitude to Dr. Raphael T. Haftka and Dr. Rodolphe Le

Riche for giving me the opportunity to complete this dual Ph.D. I thank them for the

time and effort they spent to make this joint-degree program possible. I also thank

them for their excellent scientific guidance and for their never-ending enthusiasm to

explore new areas. Working with two advisors, and being part of two teams has been

a great source of inspiration and an enriching human experience.

I would also like to thank Drs. Fortunier, Kim, Pardalos, Sankar, Schoenauer,

and Vautrin for agreeing to be members of my Ph.D. supervisory committee and for

taking the time to review this dissertation.

I am grateful for the friendship of my colleagues in the Structural and Multi-

disciplinary Optimization Research Group, Amit Kale, Melih Papila, Xueyong Qu,

Palani Ramu, Raluca Rosca, Jaco Schutte, Satchi Venkataraman, who made my stud-

ies in Florida a pleasurable experience, and of the MeM department in the Ecole des

Mines, Pierre Celle, Bernadette Degache, Sylvain Drapier, Marco Gigliotti, Jihed Je-

didi, Jerome Molimard, Joel Monnatte, Benoit Serre, Stiphane Vacher: they made

life in Saint-Etienne as colorful as in Florida.

Finally, I want to thank my family for trusting and supporting me during these

four years.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ............................... ix

LIST OF FIGURES ................................ x

LIST OF ACRONYMS ................... ....... xv

LIST OF SYMBOLS ................... ......... xvi

ABSTRACT ... .. .. .. .. ... .. .. ... .. .. .. .. ... xviii

CHAPTER

1 INTRODUCTION .................... ....... 1

1.1 Optimization of composite laminates ...... .......... 3
1.2 From genetic algorithms to estimation of distribution algorithms 7
1.3 Goals of this research .......... ............... 13

2 ESTIMATION OF DISTRIBUTION ALGORITHMS ....... .. 15

2.1 Preliminary: stochastic model of a deterministic function ..... 15
2.2 Estimation of distribution algorithms ............... .. 18
2.2.1 General principle .................. .. 18
2.2.2 Illustration .................. ........ .. 19
2.3 The general estimation of distribution algorithm . ... 20
2.4 Selection schemes .................. ...... .. 22
2.5 Estimating the distribution of promising points: theoretical issues 26
2.5.1 The conservation law for generalization performance .. 26
2.5.2 Simple example .................. .. .. .. 27
2.5.3 The bias versus variance compromise . . 28
2.5.4 Assessing the accuracy of an estimate . . 29
2.6 Estimating the distribution of promising points in practice . 31
2.7 Estimation of distribution algorithms and other stochastic algorithms 32









3 THE UNIVARIATE MARGINAL DISTRIBUTION ALGORITHM ... 35

3.1 Algorithm ............................... 35
3.2 Study of the original UMDA .......... ....... .... 37
3.2.1 Problem description .......... ........... 38
3.2.2 Population size and selection pressure ............ 39
3.2.3 Dimensionality . . ........ .... 46
3.3 Investigation of possible modifications to UMDA: memory and mu-
tation ............. ... ................ 58
3.3.1 Memory ............... ........ .. 58
3.3.2 M utation .. ......... . .... ...... 60
3.3.3 Bound on the probabilities ................. .. 63
3.3.4 Elitism . . . . . . ... .. 64
3.3.5 Conclusion of the parameter study . . ..... 66
3.4 Comparison on three problems ................... .. 66
3.4.1 Presentation of the algorithms . . ..... 67
3.4.2 Constrained maximization of the first natural frequency .67
3.4.3 Minimize A66 . .. . . . 72
3.4.4 Strength maximization ............ .. .. .. 74
3.5 Conclusion .................. ............ .. 76

4 THE DOUBLE-DISTRIBUTION OPTIMIZATION ALGORITHM ... 77

4.1 M otivation .................. ............ .. 77
4.2 Principles ................... . ..... 80
4.2.1 Identification of dependencies and data simplification .. 80
4.2.2 Sampling from two distributions .............. .. 82
4.3 The double-distribution optimization algorithm . ... 87
4.3.1 General algorithm .................. .. 87
4.4 Application to composite optimization .............. .. 88
4.4.1 DDOA for composites ... . . . 88
4.4.2 Issues associated with the representation of the auxiliary
variable distribution ................... .. 91
4.4.3 2D didactic example .................. ..... 93
4.4.4 Extensional problem. .................. .... 94
4.4.5 Extensional-flexural problem ............. ..100
4.4.6 Strength problem . . . .... ... 104
4.4.7 A comparison of diversity in UMDA and DDOA ...... 106
4.5 Performance with optimized parameters . . 115
4.5.1 Parameter study and best setting . . 115
4.6 Generalization of DDOA to continuous optimization problem 119
4.6.1 Problem description ................ . 119
4.6.2 The algorithms ................. .... 120
4.6.3 Results and discussion .. ............. ... .. 121
4.6.4 Improvement to the algorithm . . .... 125









4.7 Conclusion .................. ............ 127

5 CONCLUSION .................. ........... .. 129

5.1 Summary of the findings .................. .... 129
5.1.1 Incorporation of variable dependencies through physics-based
auxiliary variables ............. .... . 130
5.1.2 Control of the diversity ...... . . .. 131
5.1.3 Combination of distribution-based and directional search mech-
anisms ...... .. .... .. .... 132
5.2 Potential avenues for future research . . .... 133

6 RESUME ................... ................ 134

6.1 Optimisation de stratifi6s composites . . 134
6.2 Optimisation statistique ................ ... 135
6.3 Contenu de la thise ............... .... .. 139
6.4 Conclusions ............... .......... 144

REFERENCES .. ............... ............. ..146

APPENDICES

A COMPOSITE LAMINATE OPTIMIZATION . . 153

A.1 Basic mechanics of laminated plates . . 153
A.1.1 Stiffness matrices ..... ........... . .. 153
A.1.2 Lamination parameters .............. .. 155
A.2 Mechanical properties used in this work . . .... 159
A.2.1 Poisson's ratio ................. ... .. 159
A.2.2 Coefficient of thermal expansion . . ..... 160
A.2.3 First natural frequency .............. .. .. 162
A .2.4 Strength .. ... .. .. .. .. ... .. .. .. ..... 162

B BANDWIDTH SELECTION FOR DDOA . . 164

B.1 Kernel density estimation ...... .......... . 164
B.2 Determination of the bandwidth .................. 166
B.2.1 Effect of a on the accuracy ..... . . .... 166
B.2.2 Maximum likelihood method . . 169
B.3 Application in the context of DDOA . . .... 169
B.4 Effect of the bandwidth: empirical results . . 171

C LAMINATION PARAMETER PROPORTIONAL ACCEPTANCE: EX-
PERIMENTAL RESULTS ................ .. 174









D MULTIOBJECTIVE OPTIMIZATION OF LAMINATES ....... 176

D.1 Problem description ................... ...... 176
D.2 Introduction to multiobjective optimization ............ 178
D.3 Genetic algorithm ................... ..... 181
D.3.1 Laminate encoding ............ ... . 182
D.3.2 Fitness function .................. .. ... 183
D.3.3 Genetic operators ................ ... 185
D.3.4 Implementation . ....... . 188
D.4 Results ....... ......... ........ .. ....... 188
D.5 Concluding remarks .................. .. ..... 194

BIOGRAPHICAL SKETCH .................. ......... 196















LIST OF TABLES
Table page

2-1 Sample points, p = 20 .................. ...... .. .. 27

3-1 Material properties of graphite-epoxy ................. 39

3-2 Mean best fitness at the fifth generation ................ .41

3-3 Material properties of graphite-epoxy. ................ 68

3-4 Material properties of glass-epoxy. .................. 74

4-1 C'!h i o':teristics of the three algorithms . . ....... 116

D-1 Material properties of graphite-epoxy and glass-epoxy . ... 178

D-2 Parameters associated with the genetic operators . . .... 189

D-3 Optimum designs of the minimization of the composite objective functionl90















LIST OF FIGURES


Figure page

2-1 Stochastic model of a function .................. ..... 16

2-2 1D objective function .................. ...... .. .. 20

2-3 Convergence of the distribution p(x) .................. 20

2-4 General Estimation of Distribution Algorithm ............ ..22

2-5 Selection as a noisy classification task ................. ..25

2-6 Estimating the distribution from a sample .............. ..27

3-1 Basic Univariate Marginal Distribution Algorithm . .... 37

3-2 Fitness landscape All ............... ..... 39

3-3 Evolution of the mean best fitness ............. .. .. 41

3-4 Reliability of the optimization for various combinations of populations
sizes as a function of the number of iterations . .42

3-5 Reliability of the optimization for various combinations of populations
sizes as a function of the number of evaluations . . ... 43

3-6 Evolution of the probabilities for 01 -- a, All max problem . 45

3-7 Ordering convention for a balanced symmetric laminate. ...... ..48

3-8 Fitness landscape of the first natural frequency fl .......... ..48

3-9 Expected convergence time of the stochastic hill-climber for the max
All and vibration problems ................ ....... 50

3-10 Number of analyses until the average maximum fitness reaches I '-. of
the optimal fitness, max All problem ................ ..54

x









3-11 Number of analyses until the average maximum fitness reaches -' of
the optimal fitness, vibration problem ............... ..55

3-12 Evolution of the probability distributions. .. . . ...... 56

3-13 Comparison of the performance of SHC on the Max All problem and
the vibration problem. ................ .. .... 57

3-14 Effect of memory on UMDA ................ .... 59

3-15 Influence of mutation on the reliability for two UMDA variants . 62

3-16 Effect of mutation for p, = 0.005 .................. ... 63

3-17 Effect of imposing a lower bound on the probabilities on the reliability. 64

3-18 Effect of an elitist strategy on reliability. ............ 65

3-19 Mean best fitness and reliability for SHC, UMDA, and GA for the con-
strained maximization of the first natural frequency of a laminated
plate (n = 8) .................. ........... .. 70

3-20 Mean best fitness and reliability for SHC, UMDA, and GA for the con-
strained maximization of the first natural frequency of a laminated
plate (n = 15) .. .. ... .. .. .. .. ... .. .. .. ..... .. 71

3-21 Evolution of the marginal distribution p(ax), UMDA, max A66 problem. 73

3-22 Reliability and mean best fitness of UMDA and SHC for the strength
problem ............... ........... .. .. 75

4-1 Selected points and univariate distribution for the constrained vibra-
tion problem ............... ........... .. 78

4-2 Interpretation of variable dependencies as the joint action of hidden
variables ............... .............. .. 80

4-3 Incorporation of variable dependencies through auxiliary variables. .. 82

4-4 Probabilistic acceptance scheme: variance reduction . ... 84

4-5 Influence of the two-distribution scheme on the variance . ... 86

4-6 Flowchart of the double-distribution optimization algorithm. . 87

xi









4-7 Effect of the two-distribution sampling procedure .......... ..93

4-8 Fitness landscape of the penalized A22 for n = 2 .... ....... 95

4-9 Mean maximum fitness and reliability for the constrained max A22
problem, n = 6 (50 runs). .................. .... 98

4-10 Mean maximum fitness and reliability for the constrained max A22
problem, n = 12 (50 runs). .................. .... 99

4-11 Fitness landscape of the penalized CTE problem for n = 2. ..... 101

4-12 Mean maximum fitness and reliability for the constrained min CTE
problem n = 6. .................. ...... 102

4-13 Mean maximum fitness and reliability for the constrained min CTE
problem, n = 12 (50 runs). .......... .... 103

4-14 Mean best fitness and reliability for the strength problem, n = 12 (50
runs). .......... .. ....... .... 106

4-15 Diversity in the criterion domain for UMDA and DDOA. ...... ..109

4-16 Diversity in the variable domain for UMDA and DDOA. ...... ..110

4-17 Distribution of the fitness of the best solution at each iteration for
UMDA and DDOA. .................. ..... 111

4-18 Effect of mutation for UMDA and DDOA . . ...... 113

4-19 Effect of bounds on the probability distributions p(0k) for UMDA and
DDOA ............................... ...... 114

4-20 Compared performances of the optimized GA, UMDA, and DDOA. 117

4-21 Best schemes for seven population sizes, for GA, UMDA, and DDOA. 118

4-22 Contours of the penalized fitness function for n 2 . ... 120

4-23 Evolution of the best fitness for cUMDA and cDDOA, n = 2 (average
over 50 runs). .................. .. ...... 122

4-24 Convergence of the primary and auxiliary distributions for cDDOA,
n = 2 (50 runs) .. ... .. .. .. .. ... .. .. ..... 123

xii









4-25 Evolution of the best fitness for cUMDA and cDDOA . ... 124

4-26 Effect of an insufficient coverage of the desing space . ... 125

4-27 Evolution of the best fitness for cUMDA and the modified cDDOA 126

6-1 Principe des algorithmes a estimation de distribution . ... 136

6-2 Algorithme a deux distributions DDOA ................ ..142

A-1 Ply-numbering convention for balanced symmetric laminates ..... 157

A-2 Feasible domain for pairs of lamination parameters . ... 158

A-3 Simply-supported rectangular plate .................. 162

B-1 Effect of the bandwidth .................. ....... 165

B-2 Accurate estimate of the distribution p(V,, V*) of selected points for
the constrained maximization of A22. ................ 167

B-3 Influence of the bandwidth a on the approximation error for a sample
size of p = 50 points. .................. ..... 168

B-4 Example of an approximation A(V,*, V*) obtained for p = 50 points
and a = 0.18. .. .. ... .. .. .. .. ... .. .... .. 168

B-5 Log likelihood L"1 as a function of a for n = 6, p = 30, constrained
maximization of A22 .................. ..... .. 171

B-6 Effect of the target point creation procedure on the best fitness 172

B-7 Influence of the boundaries on the search distribution: "reject" and
"no reject" target point creation schemes . . 173

C-1 Effect of a lamination parameter probability proportional acceptance
scheme on the best fitness .................. ...... 175

D-1 Genetic algorithm .................. .......... 182

D-2 Two-point crossover .................. ....... .. 186

D-3 Simple mutation .................. ........... 187

D-4 Permutation .................. ............. 187

xiii









D-5 Addition and deletion .................. ........ .. 188

D-6 Sandwich structure of the optimum design . . ..... 190

D-7 Pareto curve for the minimum weight and cost design of the laminated
plate ................... ... ........ 192

D-8 Limitations of the weighting method for constructing a concave Pareto
front . . . . . . . .. 193















LIST OF ACRONYMS
DDOA double-distribution optimization algorithm

EA evolutionary algorithm

EDA estimation of distribution algorithm

ES evolution strategy

GA genetic algorithm

KDE kernel density estimate

PBIL probability-based incremental learning

UMDA univariate marginal distribution algorithm















LIST OF SYMBOLS



Optimization problem definition


(X1iX2, .. ,Xn)


design variables

number of design variables, problem dimension

objective function, fitness function


Estimation of distribution algorithms


p8(x)

p(x)

A






T








v -(Vi, ,..., )
m



rn


selection probability

probability distribution of selected points

population size

number of selected points

candidate pool size

selection ratio for truncation selection

mutation probability

bound on marginal probabilities

bandwidth of the kernel density estimate

auxiliary variables

number of auxiliary variables









Composite laminates

0 (01, 02 ... ,n)




[01/ 02/... /0n











V* V3 1 W W3


laminate ply angles with respect to a reference coordi-

nate system

balanced symmetric stacking sequence: Ok stands for

the two-ply stack Ok/ k. The 's' subscript indicates

that only the top half of the laminate is given: the

lower half is obtained by symmetry with respect to the

mid-plane.

lamination parameters: contribution of the geometry to

the laminate stiffness


Evolutionary computation terminolgy

fitness measure of the goodness of a candidate solution

function

population set of candidate solutions to the problem

generation iteration

chromosome candidate solution

gene design variable (in this work)


xvii


objective















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Phil. 1hi,

OPTIMIZATION OF COMPOSITE STRUCTURES
BY ESTIMATION OF DISTRIBUTION ALGORITHMS

By

Laurent Grosset

December 2004

C'!h In: Raphael T. Haftka
Major Department: Department of Mechanical and Aerospace Engineering


The design of high performance composite laminates, such as those used in

aerospace structures, leads to complex combinatorial optimization problems that can-

not be addressed by conventional methods. These problems are typically solved by

stochastic algorithms, such as evolutionary algorithms.


This dissertation proposes a new evolutionary algorithm for composite laminate

optimization, named Double-Distribution Optimization Algorithm (DDOA). DDOA

belongs to the family of estimation of distributions algorithms (EDA) that build a

statistical model of promising regions of the design space based on sets of good points,

and use it to guide the search. A generic framework for introducing statistical variable

dependencies by making use of the physics of the problem is proposed. The algorithm

uses two distributions simultaneously: the marginal distributions of the design vari-

ables, complemented by the distribution of auxiliary variables. The combination of

xviii









the two generates complex distributions at a low computational cost.

The dissertation demonstrates the efficiency of DDOA for several laminate opti-

mization problems where the design variables are the fiber angles and the auxiliary

variables are the lamination parameters. The results show that its reliability in

finding the optima is greater than that of a simple EDA and of a standard genetic al-

gorithm, and that its advantage increases with the problem dimension. A continuous

version of the algorithm is presented and applied to a constrained quadratic problem.

Finally, a modification of the algorithm incorporating probabilistic and directional

search mechanisms is proposed. The algorithm exhibits a faster convergence to the

optimum and opens the way for a unified framework for stochastic and directional

optimization.















CHAPTER 1
INTRODUCTION

Tod- i composite materials are essential in the development of high perfor-

mance structures. In particular, they have found usage in aerospace, automotive,

marine, civil, and sport equipment applications where their high stiffness-to-weight

or strength-to-weight, as well as their amenability to tailoring are greatly appreciated.

However they are costly and careful design is critical in order to make a rational use

of these materials: the freedom that they give to the designer comes at the price of a

more complex design process because the boundary between materials and structures

is blurred. As a result, off-the-shelf optimization methods (see Pardalos and Resende,

2002, for a survey of optimization methods) are either computationally too expensive

or not applicable to many composite design problems. The challenge of optimizing

composite structures has been the object of intensive research for many years, and

has resulted in a number of effective tools (Venkataraman and Haftka, 1999).

One of the most successful developments of the last decade in the field of lam-

inate optimization has been the introduction of genetic algorithms (GA). which are

search methods inspired by the interpretation of natural evolution as an optimization

process. One of the strengths of GAs is their ability to handle non-linear discrete

problems, which often arise in composite laminate optimization. One may argue that

another appeal of these methods is their ease of implementation and use. However,

while the method itself is simple, effective application to practical cases often requires

1







2

expertise from the user. Typically, a number of parameters have to be adjusted by

hand, based on the user's knowledge of the problem and experience with the algo-

rithm. In addition, GAs typically require a large number of function evaluations,

which can be prohibitive when a single evaluation takes minutes or hours. More

recently, several attempts have been made to reformulate genetic algorithms in a sta-

tistically more general form, giving rise to a family of optimization methods called

estimation of distribution algorithms (EDA). The main benefit of abandoning the

biological analogy is the increased control over the search it provides. Unlike GAs, in

which promising designs are created by combining features from good individuals at

random, EDAs attempt to explicitly identify the features that make these individuals

good, and to use this knowledge to create new individuals that are likely to have a

high fitness.

However there are only few examples of applications of EDA methods to engi-

neering problems, and to our knowledge, these advances have yet not reached the

field of laminate optimization. The main objective of the present work is to develop

efficient estimation of distribution algorithms for composite laminate optimization.

Our approach is dictated by one of the conclusions of the No Free Lunch theorem

(Wolpert and Macready, 1995), which expresses the fact that if an algorithm is more

efficient than some other algorithm on one class of problems, the price to p' i, is lower

performance on other classes of problems. The authors formulate the task of devel-

oping a new algorithm as follows: "The only important question is, "How do I find

good solutions for my given cost function f?" The proper answer to this question is

to start with the given f, determine certain salient features of it, and then construct







3

a search algorithm, a, specifically tailored to match those features." Consequently,

instead of creating a black box algorithm that works well for most problems, we use

our knowledge of the particular class of problems under consideration to improve

performance. We will show that by incorporating information about the physics of

the problem via carefully chosen auxiliary variables, one can significantly improve the

efficiency of the search.


1.1 Optimization of composite laminates


The mechanical properties of composite materials can be tailored to the problem

at hand. This constitutes an advantage but it makes the design of the structure more

complex because it involves not only the choice of the materials and of the geometry,

but also the internal arrangement of the material.

Laminated composites are made up of a stack of plies composed of stiff fibers

oriented along a given direction embedded in matrix, such as polymer resin. The

macroscopic properties of the laminate depend on the type and the geometry of the

fibers (carbon, glass, Kevlar, etc), on the matrix properties, on the fiber volume frac-

tion, on the number of plies and their thickness, and on the orientation of the fibers.

For unidirectional composites, the strength and the stiffness are considerably higher

in the direction of the fibers than in the transverse direction. The purpose of opti-

mization is to determine the number of plies, their thickness as well as their material

and orientation so as to extremize certain criterion, for instance the displacement at

one point, the weight, the cost, the first natural frequency, etc., subject to constraints

on strength or manufacturing. This is known as stacking-sequence optimization.







4

In general, the variables involved in laminated composite design are mixed, i.e.

defined over discrete and continuous spaces: some are discrete by definition (the ma-

terial has to be chosen from a catalog), some become discrete due to manufacturing

constraints. In particular, in many applications, the orientation of the plies is not

arbitrary but can take only a limited set of discrete values, for example 0, 450, and

900. Consequently many composite optimization problems are combinatorial prob-

lems, notorious for being difficult to solve by traditional (gradient-based) methods.

In addition, these problems, when expressed in terms of the ply angles as design

variables, have been shown by Pedersen (1987) to possess several local optima. To

summarize, composite laminate optimization problems are in general multi-modal

discrete optimization problems.

Over the years, extensive research has been devoted to solving these problems.

The high computational cost of available discrete optimization algorithms led early

researchers to relax the discreteness constraint in order to apply readily available

methods to stacking-sequence optimization problems. In a design for minimum weight

under buckling constraints, Schmit and Farshi (1977) forced discreteness of the ori-

entations by using the total thickness of each of some prespecified angles as design

variables. These were treated as continuous during the optimization and the final

result was rounded to the closest multiple of the ply thickness available from the

manufacturer. The method was not capable of optimizing the stacking sequence,

furthermore, the rounding procedure did not guarantee optimality. The difficulty

of dealing with discrete ply angles was addressed by Shin et al. (1990) who used a

gradually increasing penalty approach which forced a continuous search algorithm to







5

converge to discrete values for the ply orientations. The main difficulty was to choose

an appropriate adaptation strategy for the penalty parameter in order to prevent both

premature convergence to a local optimum and convergence to non-discrete solutions.

The second undesirable characteristics of stacking-sequence optimization problems,

namely the non-convexity was removed by Haftka and Walsh (1992) for the buckling

load maximization problem. They transformed the non-linear optimization problem

into a linear programming problem by formulating it in terms of "ply-orientation-

idP ii Il y design variables. The resulting problem was then amenable to very efficient

linear programming algorithms. However, the method is limited to stiffness design

and cannot address strength constraints. More recently, Foldager et al. (1998) used

the fact that many laminate optimization problems are convex when expressed in

terms of higher-level mechanical quantities called lamination parameters. They per-

formed a continuous optimization using ply-angles and lamination parameters simul-

taneously with the aim of forcing convexity. They used a sequential procedure, where

ever improving starting points were chosen based on lamination parameter sensitivity

information.

In the early 1990's, stochastic search methods began to be applied to stacking-

sequence optimization because they exhibit several desirable attributes: firstly, they

do not require continuity or gradient information and can therefore be applied to in-

teger or discrete problems. Secondly, they are inherently global and are consequently

less likely to get trapped in local optima. Lombardi et al. (1992) solved the bucking

load maximization by using simulated annealing. The algorithm demonstrated high

reliability in finding near-optimal designs. Callahan and Weeks (1992) first applied







6

a genetic algorithm to solve a stacking-sequence problem by using the ply orienta-

tions as discrete design variables. Le Riche and Haftka (1993) developed a genetic

algorithm that took advantage of the physics of the problem by devising genetic

operators, such as the permutation operator, specifically targeted at modifying cer-

tain mechanical quantities, such as bending stiffness while leaving other quantities

constant. They were able to improve efficiency because the number of unsuccessful

mutations could be reduced. An alternate strategic was proposed by Autio (2000)

to incorporate physics-based knowledge into a the optimization: first a continuous

optimization is applied, with the lamination parameters as design variables, then a

genetic algorithm is used to determine the stacking-sequence that yields the optimal

lamination parameters. Other researchers have reported successful applications of

stochastic optimization approaches to laminate optimization problems. For instance

Zabinsky (1998) advocated the use of simple random search algorithms called Im-

proving Hit-and-Run (IHR) in combination with "black box" numerical simulation

tools because they provide both efficient and robust methods.

During the last decade, genetic algorithms have become a standard tool for

stacking-sequence optimization (e.g. Punch et al., 1995; McMahon et al., 1998; Park

et al., 2003). At the same time, the field of genetic optimization has undergone ma-

jor mutations. Genetic algorithms, and more generally evolutionary algorithms have

been thoroughly studied and improved. That research has given rise to a new type

of genetic algorithms based on statistical methods. The next section explains how

the transition from standard GAs to search distribution evolutionary algorithms can

potentially be beneficial to stacking sequence optimization.







7

1.2 From genetic algorithms to estimation of distribution algorithms


Genetic algorithms (GA) (Holland, 1975; Goldberg, 1989) are inspired from the

biological metaphor of the Darwinian theory of evolution, which states that a species

is capable of adapting to a changing environment because those individuals of the

population that are best adapted are more likely to have descendants and therefore

to pass on their good traits to the next generation. The underlying assumptions

are that natural selection produces individuals that are optimal with respect to their

environment and that one can duplicate the process on an artificial optimization

problem with a computer.


Genetic algorithms start by initializing a population of individuals at random.

Each individual represents a possible configuration of the structure to optimize, coded

into a string of finite length, called chromosome. Then 1,-ulution" is simulated by

successively applying selection of parents, recombination, mutation and replacement

of the parents, creating a sequence of populations of increasingly "fit" individuals

(individuals with high objective function evaluation). Recombination, or crossover,

is an operator that swaps genetic material between two or more parent individuals

to create children. Mutation introduces random perturbation in an individual, for

instance by changing the value of one gene.

GAs are part of a more general class of stochastic algorithms called evolutionary

algorithms, which also includes evolutionary programming (EP) (Fogel et al., 1966)

and evolution strategies (ES) (Rechenberg, 1973). All three algorithms are based

on the Darwinian evolution principle. Although initially these methods had strong







8

differences, i .--, 1s, the frontiers have blurred, and they differ only in implemen-

tation details. Mostly, GAs typically use binary representation and use crossover as

the main operator and mutation as a background operator, EP uses representations

tailored to the problem and does not use crossover, and ES's work mainly with con-

tinuous representation. While EP and ES use both crossover and mutation, they rely

heavily on mutation, which is usually adaptive. For a more detailed introduction to

evolutionary algorithms, the reader is referred to Spears et al. (1993) or Back (1996).

Although genetic algorithms usually designate evolutionary algorithms that use bi-

nary representation, we will use the expression in a broader sense to refer to discrete

evolutionary algorithms, even when they work with non-binary variables and make

ample use of mutation.

Since their inception, genetic algorithms have been the subject of intensive inves-

tigation to understand how they work and the types of problems on which they can

be expected to perform well (relative to other algorithms). Holland (1975) introduced

the concept of schemata, or partially defined strings to analyze the performance of

GAs. This led to the Building Block Hypothesis, which states that "GA works well

when instances of low-order, short schemas ('building blocks') that confer high fit-

ness can be recombined to form instances of schemas that confer even higher fitness"

(Mil\ chell et al., 1994). In other words, the efficiency increases when genes that work

together can be detected by partially sampling the design space and their association

has a high probability of being preserved during GA processing. A reflection on v---

to detect and preserve linkage between genes resulted in the emergence of a new class

of GAs. Initial work relied on reordering operator to group these genes, so that they







9

do not get parted by crossover,1 but Goldberg and Bridges (1990) showed that these

operators were too slow to be effective. The messy genetic algorithm (mGA, Gold-

berg et al., 1989) was an attempt to "1. I i the structure of the problem by using a

flexible coding that enables the algorithm itself to evolve the .,1i i'.ency relation be-

tween genes so as to keep strongly correlated variables together. By minimizing the

defining length of these building blocks, the chance of crossover breaking it would be

minimized. This line of research has led to the conclusion that the power of natural

evolution was bounded and that linkage could only be efficiently learned by explicit

statistical methods (Harik, 1999).

At the other end of the spectrum, the increasing complexity of algorithms pro-

voked skepticism among certain researchers, such as B Iuii who claimed that one

could dispense with the population (B i1ji i 1994; B 1i i and Caruana, 1995). Based

on the hypothesis that the role of the population is to keep memory of good regions

of the design space, he proposed to replace it with an explicit probability vector

that summarizes information collected from all the points visited so far during the

search. That vector is used to generate new individuals with high probability of

having high fitness function evaluations. The new observations are in turn used to

update the model by moving the probabilities toward the best individual. The result-

ing algorithm is called PBIL, for probability-based incremental learning. Extensive

comparisons with a simple GA, which has a higher level of sophistication revealed




1 In standard crossover operators, genes that lie close to each other are less likely
to be separated than genes that are situated far apart on the chromosome.







10

that PBIL outperforms GA on many standard test problems (see also B il1i (1993)).

Finally, one last endeavor contributed to the emergence of estimation of distri-

bution algorithms: the attempt to study GA's behavior by Markov chain analysis

to obtain convergence time estimates (Nix and Vose, 1991). These methods see evo-

lutionary algorithms as dynamical systems and try to determine their .i-vmptotic

behavior (stability, convergence time, etc). They involve modeling the population

in terms of transition probabilities. Naturally, researchers of that field realized that

behaviors similar to GA's could be achieved by replacing populations by probability

distributions. For example, Miihlenbein and Mahnig (2000) showed that the tradi-

tional crossover operator can be replaced by a probabilistic model that represents the

distribution of good designs under certain conditions.

All three approaches converged toward the emergence of estimation of distri-

bution algorithms. These algorithms use a probabilistic model of good individuals

encountered during the optimization to guide the search toward promising regions of

the design space. Most of the research conducted in the field of EDAs is devoted to

the construction of good probabilistic models.

Miihlenbein and Mahnig (2000) found that without selection, the gene frequen-

cies of a population converged to a state where genes become decorrelated from each

other. Based on this conclusion, they created an algorithm that explicitly stores the

univariate allele frequencies in the form of a probability vector. The algorithm, called

univariate marginal distribution algorithm (UMDA) is very similar to the PBIL algo-

rithm. Harik et al. (1999) developed a similar probabilistic algorithm named compact

genetic algorithm (cGA), which mostly differs in its probability update rule.







11

All of these algorithms used simple probabilistic models that do not account

for variable interactions. In other words, they assume that the variables have no

combined action (independent variables) and monitor only univariate marginal fre-

quencies. In many practical situations, however, variables do influence one another.

For instance, in the knapsack problem,2 the size of the items already chosen affects

the choice of the remaining items, or in stacking sequence design for maximum buck-

ling load, some groups of plies can determine the buckling mode, and alter the most

desirable angle for other plies. Therefore it is important to account for interaction

between variables. A practical way of modeling variable interaction in the probabilis-

tic sense is to use B li-, -i in networks. B li- -i i, networks are made up of a structure

(directed .,. i-, 1 graph), which indicates the variables that are interdependent, and

numbers (conditional probabilities), which express the strength of the relationship.

Although in theory it is possible to model any order of interaction between variables,

the number of model parameters (the conditional probabilities associated with each

link of the graph) to estimate from the data, and hence the size of the sample required

to obtain an accurate graph increases very rapidly with the order of the interactions

considered. In addition, having a high order model may be harmful, generating only

duplicates of the population being modeled: this is especially detrimental when the

regions already visited do not contain the optimum, because the model will prevent




2 Task of selecting items to be packed into a knapsack. The items are chosen from
a large number of objects with different volumes. One can choose as many items of
whatever volume to go into the knapsack as one likes, provided that the total volume
of the knapsack is filled.







12

the exploration of new areas, as will be seen in Section 4.6.

The simplest type of interactions are pairwise interactions, which relate two vari-

ables (genes). Several approaches have been proposed to incorporate pairwise inter-

actions into the probabilistic model, depending on what measure of coupling is used,

and how the model is constructed. Even though in general an arbitrary joint dis-

tribution cannot be expressed in terms of pairwise conditional probabilities, the first

order interaction model can be fitted to the distribution in the sense of some distance.

De Bonet et al. (1997) used a chain model to represent the joint distribution of promis-

ing individuals. Their algorithm, called MIMIC (for mutual information maximizing

input clustering) looks for the gene permutation and the conditional probabilities that

minimize the Kullback-Lieber distance with respect to the true distribution of the

selected individuals. They applied MIMIC algorithm to solve several deceptive prob-

lems (problems where univariate frequencies alone drive the optimization away from

the optimum) and observed considerable gain over GA and PBIL. B i1li I and Davies

(1997) extended MIMIC algorithm by using dep-dl -i, -tree instead of chain model,

thus offering more flexibility in the representation of joint distributions. Even higher

order interactions can be represented by using B li, -i ,i Networks. Pelikan et al.

(1999) developed an algorithm, called B li-, -i ,i Optimization Algorithm (BOA) that

can model any order of interaction between variables. Experimental results on sev-

eral test problems indicate that BOA benefits from learning second order interactions

on functions involving strong variable interactions, while simple GAs or zeroth order

probabilistic algorithms are more effective on problems that do not exhibit strong

dependencies.







13

To date, there have been only few applications of EDAs to engineering problems.

The only application that we found was a comparison of the relative performances of

GA and PBIL for two combinatorial problems: a turbine balancing problem and a

stock cutting problem (Carter, 1997). Carter found that GA performed significantly

better than PBIL on both problems. One possible explanation for the better perfor-

mance of GA on that problem may be that it effectively uses a swap operator, which

exchanges two genes at random, which cannot be obtained with the PBIL algorithm.


1.3 Goals of this research


The present work aims at developing efficient strategies for stacking sequence op-

timization through the implementation and improvement of estimation of distribution

algorithms. We start by implementing simple EDAs to solve laminate optimization

problems, with only minor changes to the original UMDA algorithm. Then modifica-

tions are proposed to improve the performance. In particular, the main emphasis of

this work is the incorporation of physics-based knowledge to the algorithm to improve

the accuracy of the statistical model of promising regions, and hence the efficiency of

the search. This dissertation is organized as follows:


in C'! lpter 2, a general introduction to EDAs is provided. General principles

are presented, along with the inherent difficulties associated with the estimation

of distributions;

C'! lpter 3 presents a simple EDA implementation and its application to com-

posite laminate optimization. The algorithm's performance is investigated, and

improvements are proposed.







14

* in C'!i ipter 4, physics-based knowledge is used to improve the performance of

the algorithm introduced in Chapter 3. This is achieved by monitoring the

probability distributions of good individuals both in the ply angle space and

in the lamination parameter space to obtain complementary information about

promising regions. The goal is to model variable interactions with two simple

statistical distributions.

* C('!i lter 5 provides a summary of the dissertation and identifies future areas of

research.

* C('!i lter 6 is a summary of this work in French, in compliance with the joint

degree agreement between the University of Florida and the Ecole des Mines

de Saint-Etienne.















CHAPTER 2
ESTIMATION OF DISTRIBUTION ALGORITHMS


In this chapter, a general introduction to estimation of distribution algorithms

(EDAs) is provided. Estimation of distribution algorithms search the design space

by iteratively estimating the distribution of promising points and using it to guide

the search in subsequent generations. One goal of this chapter is to approach EDAs

from a statistical standpoint in order to identify difficulties and outline potential

avenues to improve their efficiency. After introducing a conceptual EDA dealing with

probability distributions only, the actual algorithm is obtained by estimating these

distributions from samples of points.


2.1 Preliminary: stochastic model of a deterministic function


Let F(x) be a deterministic function to be maximized over a domain D. The goal

of global optimization is to find the set of maxima 0 = {x c D : Vy E D, F(y) <

F(x)} (see Horst and Pardalos, 1995, for a survey of global optimization methods).

When prior information about the form of F is available (for instance if we know

that F is linear, or convex over D), then its value at some points of the design space

can be extrapolated to other points of D, and information about the location of the

optimum can be extracted. However, when no prior information about the form

of the objective function is available, assumptions need to be made. A convenient

way to express uncertainty about the true form of F is to define a prior distribution

15







16

over a set of functions,1 and update that distribution on the basis of incoming

observations, thereby reducing the uncertainty about F. Even if F is deterministic,

it will be represented as a probability distribution reflecting our lack of knowledge,

as illustrated conceptually in Figure 2-1.


5 5











0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
x x
(a) Functions compatible with the observa- (b) Contours of p(F|x). The combination of a
tions. prior probability about the form of F(x) and
the observations yields a distributions over D.

Figure 2-1: Representation of our uncertainty about the objective function as a
probability distribution p(Flx).



This representation of uncertainties about the objective function is explicitly used

in global optimization algorithms called stochastic optimization (SO) algorithms2

(see for example Betr6 (1990); Schoen (1990), and Torn and Zilinskas (1987, C'!h i

6) for a general introduction). Given a stochastic model of F, SO define heuristics to

choose the next point to be evaluated so as to extremize some criterion. Two common




1 Often, this prior distribution is only implicit, in the form of a bias toward simple,
smooth functions.

2 Sometimes, these algorithms are referred to as B li, -i i Algorithms, because they
use B li- -' rule to update probabilities when new observations are made.









criteria are:


1. maximize the probability that F exceeds some value Fo;

2. maximize the expected value of F.


Estimation of distribution algorithms (EDAs) derive from this stochastic repre-

sentation of the objective function. They focus on the set of sood points" S (for

example the set of points whose fitness exceeds Fo). The stochastic model p(Flx),

constructed from a set of observations xi E S, i 1,..., N, combined with the def-

inition of the set of promising points S determines the probability of F being in S,

denoted by p(F E Six). The probability distribution p(xlF C S) is then obtained by

B ,v -' rule:

p(xJF c S) p(F E Slx)p(x)
p(x|F E S) = (2.1)
p(F c S)

where p(F e S) is a normalizing constant, and the prior probability p(x) reflects

a potential bias toward certain regions of the space. When no region is preferred

a priori, the probability density of promising regions p(x F E S) is equal to the

probability that the value of F belongs to S.


Our uncertainty about the form of F gives rise to the probabilistic model of

promising regions, which dictates a probabilistic search strategy implemented in es-

timation of distribution algorithms. The idea of EDAs is to directly model the set of

promising points S (unlike SO, which models the function F) and sample from it to

obtain more good points, and eventually find the optimum. The general principles of

EDAs will be provided in the next sections.







18

2.2 Estimation of distribution algorithms

2.2.1 General principle

Let F(x) be a (fitness) function to be maximized over a domain D. Let 0

designate the set of optima 0 = {x E D : F(x) = FPt}, where Fopt is the maximum

if F over D. The principle of estimation of distribution algorithms is to construct a

sequence of distributions {pt(x)}t,, that converges toward the uniform distribution

over the set of global optima 0.

The sequence of distributions is obtained as follows:

1. p(x) is uniform over D

2. pt+l(x) is the distribution over D obtained after application of a fitness-based

selection operator to all the points of the domain. The selection operator is

defined by its probability p'(x) E [0, 1], which is the probability of accepting a

point x in an acceptance/rejection process. Using B i, -' rule, the recurrence

equation reads

t+1 Pt(x)pS(x)
pt+ (x) (2.2)
p8

where p8 = f pt(x)p'(x)dx is a normalizing constant.

For the sequence to converge to the uniform distribution over 0, two conditions

are required:

1. F(x,) = F(xj) = ps(xi) = ps(xj)

(the selection probabilities of two points having the same fitness function eval-

uation are the same),

2. let Dt {x D : pt(x) > 0}, let Rt {F(x) : x e D},









Vt > 0, FOc R7 such that

F(x) < Fo
F ps(x) < p(x,) for (x,,xj) c (D')2

F(xj) > Fo'
(the probability of low-fitness points is smaller than that of high-fitness points)

It has been shown (Berny, 1999, for a discrete case) that the sequence of prob-

ability distributions pt(x) generated by the successive applications of fitness-based

selection to the whole space is identical to the sequence of distributions that would

be obtained by performing a gradient-based maximization of the expectation of the

fitness

Ep[F] j= F(x)p(x)dx

with respect to the distribution p(x) in the space of the probabilities p.

2.2.2 Illustration


The evolution of the distribution of promising points is illustrated for the simple

ID non-negative function:


F(x) =50- 60x2 + 50x5 + 20x10 + 20 sin(40x/r) (2.3)


The function has three local optima on [0, 1], at x = 0.11, x = 0.61, and x = 1.00, as

shown in Figure 2-2.

Given a fitness function and a selection operator, it is possible to explicitly

determine the distribution of promising points at any iteration t. For instance, when

fitness proportional selection is used, the selection probability is given by


p(x)= p(Sx) af(x) ,








20


where a can be chosen arbitrarily. The probability distribution evolves according to



pt+(x) p(xlS)

p(Slx)p(x)

J p(S x)p(x)dx


f(x)pt(x)

The distribution at times t = 0, t = 5, t = 10, and t = 20 is shown in Figure 2-3.


Starting from a uniform distribution over [0, 1], pt(x) gradually focuses on high-fitness


regions, and degenerates into a Dirac function centered about the (here unique) global


optimum.


40 -

30 -


10

0 01 02 03 04 05 06 07 08 09 1
x

Figure 2-2: The 1D objective function
has three local maxima at x = 0.11, x =
0.61, and x = 1.00, but a single global
maximum at x = 0.11.


t=0
--- t=5
12 t=10
..-- t= 20
ii t
10

8 i

6 I i

I \I
4 '

2 / '

0 01 02 03 04 05 06 07 08 09 1
x

Figure 2-3: Convergence of the distri-
bution p(x) toward the global optimum
x 0.11


2.3 The general estimation of distribution algorithm


Estimation of distribution algorithms construct a sequence of probability dis-


tributions {pt(x)}teN that converges to a uniform distribution over the set of global


optima. However, explicitly calculating the distributions using formulas such as Eq.


(2.2) is not reasonable, as is implies computing the fitness function everywhere, which









renders the whole optimization useless.

In practice, the distributions pt+(x) are not calculated explicitly, but approxi-

mated from a finite sample of size p (similar to the Ip rent population" in the context

of evolutionary computation), obtained by fitness-based selection among a set of A

points (the "child population" in EAs) generated from pt(x).

Since the selection probability distribution p'(x) is simulated from a finite sam-

ple, so that the second condition of Section 2.2.1 may not be satisfied, causing se-

lection error, and the distribution pt(x) of promising points at time t is estimated

based on a finite number of points, which generates estimation error (this will be

addressed in detail in the next section), the practical algorithm can only achieve an

approximate distribution pt(x), which is not identical to the theoretical distribution

pt(x) in general. The actual trajectory {pt(x)}tes in the space of probabilities will

be different from the theoretical trajectory. Depending on the quality of the approx-

imations employ, -l it may or may not converge to the optimum. The !i II" (as well

as the subscript t) will often be dropped in the remainder of this work for simplicity.

The general flowchart of EDAs is presented in Figure 2-4. The algorithm is

comprised of three main steps, besides the initialization, where the distribution po(x)

is chosen (when not prior knowledge about the location of the optima is available, a

uniform distribution over D is used). The first step consists in generating a population

of A points by sampling from pt(x). Then, fitness-based selection is applied to obtain

Ip ;ood" points. The new distribution pt+l(x) is finally estimated from the selected

population. The values of the population size A and the selected population size p

determine the accuracy of the approximations p'(x) and pt+l(x). The procedure is







22

INITIALIZATION
Initialization of the
distribution p(xl, 2,..., an)


SAMPLING
Creation of A points by sampling
from p(xl, 2,..., n)


SELECTION
Selection of p good points based on F


ESTIMATION
Estimation of the distribution
p(Al,x2,..., xn) of the selected points

Figure 2-4: General Estimation of Distribution Algorithm



repeated until a stopping criterion (usually a fixed number of function evaluations)

is met.


The performance of estimation of distribution algorithms is determined by two

factors:


the choice of the selection scheme (probability distribution p'(x)), which deter-

mines the bias toward fitter points of the population;

the choice of the probability distribution model, and method adopted to esti-

mate the distribution of selected points.


The next sections are devoted to these two critical components of the algorithm.


2.4 Selection schemes


In the theoretical EDA presented in Section 2.2.1, the transition from the dis-

tribution pt(x) of promising points at time t to the distribution at time t + 1 is







23

accomplished by applying selection to the whole search space D. In the actual al-

gorithm, the goal of selection is to generate p sood" points that will be used to

estimate the distribution of promising points.


A selection scheme is characterized by A, the number of points sampled from

pt(x), and the selection probability p'(xj), i = 1,..., A. The value of A determines

the departure from the theoretical selection probability (the first condition of Section

2.3 requiring that the selection probability of higher-fitness points be higher than that

of lower-fitness points may not be satisfied): larger values of A will lower the selection

error. When A is too small, high-fitness areas may be ignored, and the algorithm may

fail to locate the global optima, as will be observed in Section 4.6.


Selection determines the bias toward fitter points of the population. Let P = {al,

a2, ..., a%} designate the population of individuals sorted in decreasing order of

their fitness (F(ai+l) < F(aj)). Many different selection schemes can be devised:

fitness proportional selection, tournament selection (k points are drawn from the

population, and the point having the highest fitness evaluation is kept), Bolzmann

selection (ps(x) oc p(x)), etc. In this work, two schemes will be used:


Truncation selection: given an initial population of A individuals, the / = [LA],

-T [0, 1], individuals with the highest fitness evaluation are chosen. The selec-

tion probability distribution (without replacement) is:


1 if p'(ai) =
0 otherwise







24

Linear ranking selection: the probability of selecting an element ai of the popu-

lation is proportional to its rank i in the population. The selection probability

distribution (with replacement) is:


) 2(A- i + 1)
A(a + A 1)

The choice of a particular selection scheme determines the "selective pressure"

(also called selection intensity), or bias toward fitter points of the distribution: strong

selective pressure means that only the very best points will appear in the selected

population. Several measures of the selective pressure have been proposed: Goldberg

and Deb (1991) introduced the takeover time, or number of generations until the

best individual fills up the whole population under the sole action of selection (no

mutation and no crossover), Miihlenbein (1998) borrowed the selection intensity from

the field of breeding, Blickle and Thiele (1996) used the concept of loss of diversity

d, which is defined as the proportion of individuals that disappear in the process of

selection. The loss of diversity of various schemes was quantified by Wieczorek and

Czech (2002): they obtained values of d = 0.711 for truncation selection of ratio

7 = 0.3, and d < 0.432 for linear ranking selection (the maximum loss of diversity

is obtained for an infinite population; smaller populations achieve a lower selective

pressure).

The optimal selection scheme is problem-dependent: it is the result of a trade-off

between fast convergence to high-fitness regions, and the risk of premature conver-

gence to local optima when finite populations are used. Indeed, selection is a noisy

classification task: considering two competing candidate points xl and x2, the goal







25

is to retain the one which is more informative about promising regions. Practically,

given a partition3 of x = (xi; xji), such that F(x) = F,(xi) + Fij(xij) (such par-

titions are sometimes called I, i., i", l.,i ii, (Radcliffe, 1992) or "hyperplane" in

the EA literature), and one wants to identify good values of xI based on the observed

response F, the decision will be influenced by the value of the partition xIj. The

observed fitness F(x}) of the optimal value x} of the partition x, is in fact a random

variable F(x) = FI(xf) + FIj(xii), where xil can take arbitrary values. This gives

rise to a noisy decision task, as shown in Figure 2-5: the graph schematically shows

the distribution p(F|good) of the observed fitness of the optimal value F(x}), and

the distribution p(Flbad) of some non-optimal competing value.

0.5
0.45
0.4
0.35
p(F|bad) p(F|good)
0.3
0.25
0.2
0.15
0.1
0.05
0
0 1 2 3 4 5 6 7 8 9 10
F

Figure 2-5: Selection is a noisy classification task, as the influence of other variables
affects the fitness of a subregion.


Clearly, the decision to choose one candidate over the other will be dictated by


3 For instance, x, = (xi, X2, x3) and xii = (x4..., ,X).







26

the intrinsic goodness of the partition under consideration (means of the distribu-

tions), but also by the influence of other variables, which determines the variance of

the distributions (Ml!!, r and Goldberg, 1995). The best selection strategy directly

depends on the specific fitness configuration: when the response is dominated by one

partition, strong selection will mask the effect of other partitions, hence good values

of these might be lost. In those situations, weaker selection, which takes into account

information conveyed by poorer candidates is appropriate. If all partitions influence

the fitness equally, stronger selection can be used.


2.5 Estimating the distribution of promising points: theoretical issues


2.5.1 The conservation law for generalization performance


The estimation of a distribution is a common task in machine learning, in par-

ticular in pattern classification, and a host of techniques have been developed to

address it. A number of theoretical, as well as practical issues associated with this

task are well-known, and discussed at length in the literature. One classical prob-

lem consists in choosing a good model to approximate a distribution from a sample.

Suppose we want to estimate a distribution p(x), x E D, on the basis of p sample

points xi, i 1,... p distributed according to p(x). A fundamental result is that,

in the absence of any problem-dependent information, there is no single best approx-

imate distribution p(x) that describes the sample distribution. This results from the

No Free Lunch (NFL) Theorems (also called Conservation Law for Generalization

Performance) that prove that generalization from a finite sample of points cannot

be accomplished unless additional information about the distribution is available








27


(Wolpert, 1992; Schaffer, 1994). It will be important to be aware of these theoretical

limitations when constructing approximations to the distribution of promising points


(and of the implicit assumptions made to overcome the NFL).

2.5.2 Simple example


To illustrate the difficulty of estimating a distribution from a sample of points, let

us consider the sample of / = 20 points given in Table 2-1. These points i E [0, 1],

i = 1,... were generated according to an unknown distribution p(x), which we

seek to estimate.

Table 2-1: Sample points, p = 20

X1 X2 X1 14 X4 X 1 Xg 1g X 10 X1 X12 X 13 X 14 X15 16 17 18 li1 120
0.81 0.20 0.23 0.85 0.79 0.34 0.31 0.77 0.24 0.34 0.51 0.38 0.66 0.22 0.79 0.4 .38.387 0.70 0.55 0.63




Figure 2-6 shows the sample points, and three different possible estimated dis-

tributions p(x). Although the three distributions are very different from one another,

2
+ sample points
1.8- P1(
--. P2(X)
1.6
1. ...... P (x)
1.4 ,'


>< 1
1.2 -', ... .... % %


0.8

0.6 .'

0.4 .

0.2 -


x

Figure 2-6: Estimating the distribution from a sample. In the absence of a priori
knowledge about the distribution, one cannot discriminate between the three candi-
date approximations.







28

all three are equally valid descriptions of the sample:

the distribution pl(x) has two main modes, and several local maxima;

the distribution p2(x) has only two modes;

the distribution p3(x) considers that all the points were generated from a uni-

modal distribution, and that the apparent local density maxima result from

variability in the sample.

To discriminate between all the possible approximations to the true distribution

p(x), one has to make assumptions about its structure, or incorporate prior informa-

tion about the target function.

2.5.3 The bias versus variance compromise

Let p(x) be the target distribution to be estimated, p(xlD) the approximation

to p(x) obtained based on the data D = {xl,...,x,}. Since all the sample points

are instances of p(x), there will be random variations in D, and p(x|D) will reflect

these variations. To obtain a meaningful estimate of the error incurred when using a

particular model, one needs to average over all possible samples. The usual measure

of the error made by the approximation at point x is the root-mean-square prediction

error:

rms = ED [(p(x|D) -p(x))2] (2.5)

where ED designates the expected value with respect to sample variability.

This total error can be broken down into two components as follows:


es ED [p(x|D) p(x)]2 +ED [(p(xD) ED [p(xD)])2] (2.6)
bias2 variance







29

The first component, the expected value of the difference between the approximation

and the true value is called bias; the second component, which describes the magni-

tude of the variations about the approximation's expected value is called variance.

Both terms contribute to the total error. Bias is typically high when too simple a

model is used to approximate the distribution, it is also called modeling error. Vari-

ance results from variability in the sample points. It is also referred to as learning

error and can be reduced by increasing the sample size. The difficulty of estimating

a distribution from sample points lies in the trade-off between bias and variance, as

sophisticated models will be more flexible, and have a low bias, but be extremely

sensitive to sample variability. In contrast, the variance achieved by a simple model

will be lower, but at the cost of a larger bias.


2.5.4 Assessing the accuracy of an estimate


Even though theoretically there exists no absolute criterion for deciding which

of two approximations best describes a set of sample points when no a priori infor-

mation about the target distribution p(x) is available (NFL theorem), in practice,

heuristic methods have been proposed to choose between competing models, and to

estimate the model parameters. Those methods implicitly make regularity assump-

tions about the distribution (Schaffer, 1993): in fact, the idea of finding an optimum

over a domain without exhaustive enumeration rests on the assumption that the fit-

ness function possesses some regularity attributes that allow us to extrapolate from

previous observations, hence we restricts our attention to particular classes of func-

tions, as pointed out by Rao et al. (1995).







30

Given a set ={xi, x2,... x } of p independent identically distributed data

points and a set of competing models M/ {fmI, m2,..., mk }, three classes of methods

can be used to choose the model m that best approximates the data (Duda et al.,

2000, C'! lpter 9):


Maximum likelihood (ML) methods choose the model which maximizes the like-

lihood of the data:


m = max p(D mi)
i= 1,...,k

= max I p(xj mi)
i 1,...,k
j=1

A 1 ii r disadvantage of these methods is their tendency to overfit the data.

A variant of ML methods is the maximum a posteriori (1! AP) method, where

some models can be favored over others by assigning them a higher a priori

probability p(mi). The a posteriori probability of a model after the data has

been observed is obtained by


p(miD) p(Drn)p(i)
p(D)

where p(D) is a normalizing factor that is usually ignored. The model that

maximizes the a posteriori probability is retained.

Minimum description length methods (MLD) recast the bias-variance compro-

mise into a more general problem of finding the most compact representation of

a data set. The information theory concept of Kolmogorov complexity K (the

minimum number of bits required to represent data on a computer) is used to

measure the total complexity of the model mi and the description of the data









set in mi:

K(mi, D) = K(mi) + K(D using mi)

The model that achieves the best trade-off (minimum K(mi, D)) between the

model complexity K(mi) and the model's ability to efficiently represent the

data K(D using mi) is retained.

Cross-validation methods (CV) are classical heuristic strategies used in statis-

tics to estimate the generalization accuracy of a model. They consist in splitting

the data D into k sets Di of equal size p/k. Then k approximations are con-

structed by leaving out one set Di and estimating p(x) based on k 1 remaining

sets. Each time, a different error ci is obtained, and the generalization error

is estimated as the mean of all the chi's. A common variant of cross-validation

methods is the leave-one-out method, which is obtained for k = p.


In this work, the maximum likelihood method and the cross-validation method

will be used (i\ll is invoked implicitly in Section 3 to estimate the marginal proba-

bilities, and CV is used in Section 4 to choose the window size of the kernel density

estimate).


2.6 Estimating the distribution of promising points in practice


If there is no a priori criterion for choosing the best model, how do we go about

constructing an approximate distribution pt(x)? Based on the above considerations,

our approach will be dictated by two general rules:


Simpler is better (Occam's razor): Even though there is no problem-independent







32

theoretical reason for preferring simpler models over sophisticated ones, a prin-

ciple often invoked in machine learning is Occam's razor (overfitting avoidance),

which recommends that, for a given total error, models involving fewer param-

eters be favored. Friedman (1997) showed for classification tasks that the bias

error of simple models can be compensated by low variance error, thereby mak-

ing them more accurate than more sophisticated models (EDAs can be viewed as

classifiers that discriminate between good and bad regions of the search space).

Incorporate as much information about the problem as possible: Another way

of reducing error consists in incorporating as much information about the target

distribution as possible.

2.7 Estimation of distribution algorithms and other stochastic
algorithms

Estimation of distribution algorithms can be viewed as a generalization of evolu-

tionary algorithms (Back, 1996), and as such, they bear many similarities with genetic

algorithms (Goldberg, 1989) and evolution strategies (Rechenberg, 1973). They can

also be likened to a more recent family of stochastic optimization algorithms, called

Markov chain Monte Carlo (Gelfand and Smith, 1990).

Evolution stategies (ES) search for the optimum by (1) selecting p good points

(the parents) out of a pool of A candidate points; (2) creating new points (the

children) by applying perturbations (usually Gaussian mutation) to the selected

points. If the population of selected points is considered as a whole, the process

of choosing a parent and applying a Gaussian mutation is equivalent to sampling

from the distribution of parents, estimated by a varying window size kernel







33

method (see Appendix B). The specificity of ES lies in the strategy implemented

to select the local bandwidth (i.e. the mutation standard deviation) by an

auto-adaptive scheme, i.e. the goodness of a particular value of the bandwidth

is judged based on the the fitness of the points that are generated using that

value.

Genetic algorithms (GA) proceed by (1) selecting p good points (the parents) out

of a pool of A candidate points; (2) creating new points (the children) mainly by

applying recombination operators (crossover) to the selected points. Mutation is

applied to a small portion of the children. Viewed from a statistical standpoint,

the specific crossover operator implemented defines an implicit statistical model

for the estimation of p(x) by determining the marginal frequencies considered

for the creation of new points. For example, using a uniform crossover to

create children amounts to sampling from univariate marginal frequencies of

the parents.

Markov Chain Monte Carlo (MCMC) algorithms simulate sampling from a prob-

ability distribution p(x) by constructing a Markov chain (xt), tE N whose

invariant distribution is p(x). MC' IC use a two-step procedure to generate

new points: a set of candidate points is created based on the current state xt

and a proposal distribution q(x); then the new points are accepted or rejected

according to an acceptance probability A(xnew|xcurrent). The parallel between

conventional EDAs and MC'IlC has been drawn (giving rise to hybrid algo-

rithms called eMC'\ C, for evolutionary MC'\ C (Drugan and Thierens, 2003)):







34

EDAs can be viewed as non-homogeneous MC'\ Cs whose states are popula-

tions of p points, with proposal distribution at time t equal to the distribution

of promising points pt(x), and acceptance probability commonly set to 1, even

though other replacement strategies are possible. The idea is that after a suffi-

cient number of iterations, MC'\ IC will sample uniformly from the set of global

optima.















CHAPTER 3
THE UNIVARIATE MARGINAL DISTRIBUTION ALGORITHM

In this chapter, a simple estimation of distribution algorithm, called Univariate

Marginal Distribution Algorithm, is presented. It is based on the assumption that the

design variables are statistically independent. The basic algorithm is tested on lam-

inate optimization problems, and the influence of several parameters is investigated.

The distribution-based search mechanism is compared to a point-perturbation-based

search mechanism. Then improvements to the algorithm are proposed, and the re-

sulting algorithm is applied to more realistic laminate optimization problems.

3.1 Algorithm

Let F(x) be a function over a domain D. Our objective is to find the optima

x*. As introduced in C'! ipter 2, estimation of distribution algorithms search the

design space by estimating the distribution of promising regions. The distribution

of selected points at a given time of the optimization depends on the form of the

objective function, and on the selection procedure. In particular, it has been shown

(u\!.!i, ihleein et al., 1999) that when the objective function is a sum of contributions

from single variables,1 and Bolzmann selection is used (p'(x) oc exp( 3F(x)), where

f3 is a parameter that determines the selective pressure) the distribution of selected




1 Such functions are commonly referred to as additively decomposable functions
(ADF).









points takes the simple form


x fX2,n p(xk) (3.1)
k=1

where p(xk) is the univariate marginal distribution of variable Xk. In other words,

the variables are statistically independent, and the complete distribution can be ob-

tained from the univariate marginal distributions only. This greatly simplifies the

probability estimation process. The algorithm resulting from the variable indepen-

dence assumption, called Univariate Marginal Distribution Algorithm (\! ll11 i!lein

and Mahnig, 2000), is presented in Figure 3-1. The algorithm is identical to the gen-

eral EDA, except that the general form of p(x) is replaced by the univariate model,

thus considerably decreasing the number of parameters to be estimated, which sim-

plifies the algorithm, reduces the computational overhead required to estimate the

distribution, and decreases the uncertainty on the parameters due to variability in

the selected points. In this first variant of UMDA, no random exploration component

such as mutation is implemented, because our goal is to observed the pure statistical

search mechanisms. The potential benefits of such operators will be investigated in

Section 3.3.

In this work, we deal with discrete variables Xk (the fiber orientation of each

ply, which can take c values ai, i = 1,..., c), hence the probability functions p(xk)

are discrete probability functions. To specify p(xk), we need to know the probability

of each value ai, pki p(xk = i). Since the distribution of each variable in the

selected points follows a multinomial distribution of probability p(xk), the maximum

likelihood values of pki are simply given by the frequencies fki of each value ai in the







37

INITIALIZATION
Initialization of the distributions
P(Xl), p(X2), ..., p(Xn)


SAMPLING
Creation of A points by sampling from
P(Xl), p(X2), ..., p(Xn)


SELECTION
Selection of p good points based on F


ESTIMATION
Estimation of the distributions
P(Xl), P(X2), ..., p(Xn)
of the selected points


Figure 3-1: Basic Univariate Marginal Distribution Algorithm. The general probabil-
ity distribution of selected points is replaced by a product of univariate distributions.


population of selected points.

By incorporating information about the form of the distribution, one can obtain a

more accurate estimate of p(x). However, there is a price to p i for the simplification

achieved: if the variable independence assumption is not satisfied, the estimated

distribution will be erroneous, and the resulting algorithm may fail to locate the

optima. In the following sections, we shall apply UMDA to ADF, and study their

behavior on more complex functions.

3.2 Study of the original UMDA

UMDA, as any EDA, theoretically works with probability distributions, in prac-

tice the distributions are estimated from finite samples of p selected points. The

value of p determines the accuracy of the estimated pki: the larger the population

size, the lower the standard error on these estimated model parameters. The choice of







38

the population size will therefore will be driven by two considerations: the accuracy

of p(xk) and the computational cost one is willing to plv for it. The other factor

affecting UMDA's behavior is the selective pressure. The present section investigates

the influence of these two factors on the algorithm's performance.


3.2.1 Problem description


We first studied the convergence properties of UMDA for a problem that satisfies

the variable independence assumption. The goal is to understand the strengths and

limitations of UMDA (and more generally of EDA) in an ideal setting.


We considered the problem of maximizing the longitudinal in-plane stiffness All

of a balanced symmetric graphite-epoxy laminate2 [l/ x2/... / ]s,:


maximize All = h U + (U2 co 2xk U co 4ak) (3.2)


where the total laminate thickness was h = 0.2 in, and U1, U2, and U3 are mate-

rial invariants (see Appendix A for more details about the mechanics of composite

laminates). The material properties of graphite-epoxy are given in Table 3-1. The

design variables are the fiber angles Xk, k = 1,..., n, where Xk designates the angle

of the kth ply in the laminate with respect to a reference coordinate system. For

this problem, the fiber angles were to be chosen from {0O, 150, 300, 450, 600, 750, 900},

and the number of 0-stacks in a half-laminate was set to n = 10 (40 plies in total).




2 A laminate is symmetric if the stacking sequence is symmetric with respect to
the mid-plane, and balanced if, for each 0-degree ply, there is a -0-degree ply in the
laminate.







39

The objective function had only one maximum, at [020]s (all the fibers are aligned

with the longitudinal axis). The longitudinal in-plane stiffness of that laminate was

All = 4.38 106 lb/in. It is apparent from Equation (3.2) that the objective func-

Table 3-1: Material properties of graphite-epoxy

EL 2.18 107 psi
E2 1.38 106 psi
G12 1.55 105 psi
V12 0.26



tion belongs to the class of ADFs, hence it makes sense to use the univariate model

given in Equation (3.1) to approximate the distribution of All-based selected points.

Figure 3-2 shows the fitness landscape for n = 2.


x 10
10,













80 60 80
0 -




80 40
0 20
X1 X2

Figure 3-2: Fitness landscape Al for n = 2 (xl and x2 are in degrees).



3.2.2 Population size and selection pressure

In the basic form of UMDA, only two factors govern the algorithm's behavior:

the selection scheme, which determines ps(x) (the selection scheme encompasses the







40

selection method and the selective pressure) and the number of points p used to

estimate the distribution of promising points pt(x). The purpose of this section is

to investigate their respective influence on the algorithm's performance. UMDA was

applied to the All maximization problem. Five different selected population sizes pe

{10, 20, 50,100, 200} were tried, and truncation selection of ratios '- {0.1,0.3, 0.5}

(cf. Section 2.4) were used3


As any global optimization algorithm, an ideal estimation of distribution algo-

rithm should possess two (often conflicting) attributes, as emphasized by Back (1996,

Ch. 4) in the context of evolutionary algorithms: high convergence velocity and high

convergence reliability. In this study, the optimization reliability R was estimated as

the proportion of runs that find the optimum in a fixed number of evaluations. The

mean best fitness at each iteration was used as a measure of the convergence velocity.

Both criteria were estimated based on 50 independent runs, so that the standard

error of R was e = (R follows a binomial distribution), which is maximum

at 0.07 for R = 0.5.


Figure 3-3 shows the evolution of the mean best fitness during the optimization

for the 15 combinations of population size and selection ratio (the mean best fitness at

the fifth generation is given in Table 3-2 to help the interpretation). The graph clearly




3 This choice of p as an independent parameter (A as dependent parameter, deter-
mined by the value of 7) is unusual: in evolutionary computation, the usual approach
is to study the influence of the population size A. The rationale for this choice is to
distinguish factors that influence the nominal value of the search distribution (the
bias) and factors that affect the accuracy of the estimate (the variance).







41

shows three families of curves, corresponding to the selection ratio r: the algorithm

converges faster to high function evaluation regions for strong selection. This is not

surprising, as all the variables have the same weight in the fitness function, therefore

high-fitness points are alv--i-i more informative about good regions than lower-fitness

points. Within a family of curves (identical selection ratio), the convergence velocity


44x 106
4.4 -g j^ j- -t--

4.2- ,."_-' = 10, X= 100
-- = 10, X= 33
4 r,,,, --- -=10, X=20
2 A/y'-" --- p = 20, = 200
o 3.8 /, -- i =20, X = 66
--3.8 ro
g / i^/-- =20, = 40
S36 f -- = 50, X= 500
S3.6- / =
i= "'I -e- = 50, = 166
S51/ -o- = 50, = 100
20 3.4 i
'9 .- = 100, X= 1000
S I --- = 100, X= 333
S3.2- / .-- = 100, X =200
--- g = 200, X = 2000
3 -r
S- p = 200, = 666
--4- p = 200, = 400
2.8
0 5 10 15
number of iterations

Figure 3-3: Evolution of the mean best fitness as a function of the number of itera-
tions. The convergence velocity increases when selection becomes stronger.



increases monotonically with the sample size f, as large samples yield more accurate

estimates of the distribution of good points p(x).


Table 3-2: Mean best fitness at the fifth generation

T
0.1 0.3 0.5
10 4.33 4.14 4.05
20 4.36 4.28 4.07
p 50 4.38 4.33 4.19
100 4.38 4.36 4.24
200 N/A 4.37 4.28








42

The optimization reliability depends both on the selection ratio r and on the

selected population size /, as shown in Figure 3-4. For a given value of r, the

accuracy of the distribution p(x) is directly related to the sample size p: the larger

the value of p, the smaller the variance of the marginal frequencies fki of the selected

points.This effect is even more significant for large values of r (weak selection). When

weak selection is used, the expected value of the proportion of points that contain

the optimal value x* is barely higher than that of non optimal values. To guaranty

that the estimated value of the proportion is higher, one has to reduce the variance,

hence increase the selected population size.



f' -a-a sees~e-e~e*e*e*eSe-e-eoe
0.9 ii !
Sii =10, = 100
0.8 I P= 10, X=33
I 9 --- =10, =20
0.7- .. =20, X= 200
0= !=20, X =66
I I I
o0.6' Ii' =20,= 40
0.5 =50, X= 500
i -- =50, = 166
| 0.4- i -o- = 50, = 100
-- =100, = 1000 +
0.3 -a----X --- -- 1 =100, =333 X
0.2 / /--- g=100, =200
Mi7 -+. + ~'1 -.+. + = 200, = 2000 +
0.1 -*- = 200,= X666
S--1- p=200,X=400
0 ^ jY^f44-4-44- L
0 5 10 15 20 25 30
number of iterations

Figure 3-4: Reliability of the optimization for various combinations of populations
sizes p and A, as a function of the number of iterations. The final reliability increases
with the selected population size p.



While the evolution of the reliability as a function of the number of iterations

allows us to understand how the algorithm parameters affect the accuracy of the

distribution p(x), hence the search mechanism, the number of function evaluations







43

is a more meaningful measure of the cost associated with an optimization. Figure

3-5 shows the reliability of the 15 algorithm variants against the number of analyses.

Using that new scale, a new hierarchy between the tested schemes appears: for small


1 00 ooo*eee- -O- 0 -
r--
0.9
I I
9 i --c 4 = 10, k = 100
0.8- -' 1 =10, = 33
P- ^ = 10, k= 20
-0.7 I --- = 20, = 200
Sii -- a=20, = 66
0.6- a
6o -i = 20, = 40
0.5 -e- = 50, = 500
S5 -e- g=50, = 166
0.4- -o- =50, k= 100
.i.... .' .-- = 100, k = 1000
0.3 L = 100, k = 333
0.2- =100, =200
4-L = 200, X = 2000
0.1 t = 200, X = 666
./ -0- p=200, = 400
o jaiffwr'u'T'"^--------"----i i
0 1000 2000 3000 4000 5000 6000 7000
number of evaluations

Figure 3-5: Reliability of the optimization for various combinations of populations
sizes p and A, as a function of the number of function evaluations.



selected population sizes (p = 10 and p = 20), the best reliability is obtained for

strong selection4 while for large selected populations (p = 100 and p = 200), the

reverse order is observed. This can be explained by the fact that the variance of the

proportion of good points in the selected population is large for small numbers of




4 Note that the classical result that high selection rates require large populations
(Thierens and Goldberg, 1993) consider A as the independent parameter, consequently
increasing the selective pressure results in larger sampling errors in the selected points
and in a poorer performance. When the number of selected points p is the indepen-
dent parameter, the selection pressure can be varied without affecting the sampling
error, explaining the different conclusions presented here.







44

selected points, consequently a selective scheme has to be used to help discriminate.

In contrast, when large numbers of selected points are used, the variance is sufficiently

small to allow a good discrimination, so that the number of evaluations (A) at each

iteration determines the efficiency. This result shows the two ends of the spectrum

in EDAs: small populations lead to statistical errors, whereas algorithms using large

populations suffer from high computational costs.


This can be easily understood by examining the evolution of the probabilities

with time. Figure 3-6 shows the evolution of the probability distribution p(01), start-

ing from a uniform distribution, during the first ten iterations5 The combined effects

of selected population size p and selection ratio r are clearly visible on the graphs:

for the same selection intensity, the distribution converges more smoothly toward the

optimum value 01 = 0 (Figures (a) and (b)) with increasing p. With a small selected

population of 10 points, the distribution loses variable values: for instance variable

values 60 degrees and 90 degrees vanish between the first two iterations. In this case,

the loss of values speeds up the convergence by favoring optimal values, however, pre-

mature convergence of the probabilities can lead to the loss of optimal values, which

prevents the algorithm from finding the optimum, since no diversity injection mech-

anism (such as mutation or a lower bound on the marginal frequencies) is provided.

A larger selected population enables UMDA to obtain more accurate estimates of

all the probabilities, resulting in a more reliable optimization scheme. For a given




5 Since all variables are interchangeable, the evolution of other variables would
exhibit similar trends.









































number of iterations number of iterations

(a) A 100, p 10 (b) A 1000, p 100


1 2 3 4 5 6 7 8 9 10
number of iterations

(c) A = 200, p = 100


Figure 3-6: Evolution of the probabilities for the All max problem.







46

selected population size p = 100, Figures (b) and (c) show the faster convergence of

the distribution when stronger selection is used: p(01) converges in only 6 iterations

for = 0.1, while it has not fully converged after 10 iterations for r = 0.5.


To summarize, the selection ratio determines the convergence speed of the algo-

rithm from one generation to the next. For identical selection ratios, large selected

populations provide more accurate statistics about good alleles, thereby preventing

premature convergence. However, if a large selection pressure associated with large

selected populations constitutes the best configuration for fast convergence in terms

of iterations, the increased number of analyses required to refine the evaluation of

probabilities may result in an overall scheme which is less efficient than algorithms

based on smaller populations. This problem comes down to the issue of the allocation

of resources between exploration and exploitation: using a smaller selected popula-

tion results in inaccurate parameter estimates, and can potentially drive the search

toward poor individuals, but since fewer analyses are performed at each generation,

the loss in accuracy is compensated by the increased number of iterations permitted,

given a fixed budget.


3.2.3 Dimensionality


An important characteristic of optimization algorithm is the impact of an in-

crease of the number of variables on the time to convergence, which is called the

algorithm time complexity. In this section, we compare UMDA's time complexity to

that of a random search algorithm called Stochastic Hill-C('!ii ,I i (SHC). SHC is one

of the simplest stochastic algorithms, and has been proposed as a baseline method







47

for evaluating evolutionary algorithms' performance (Juels and Wattenberg, 1995).

The algorithms are applied to two problems: the maximization of the in-plane longi-

tudinal stiffness introduced in Section 3.2.1, and the maximization of the first natural

frequency of a simply supported rectangular graphite-epoxy laminated plate.


Problem description: frequency maximization


The first natural frequency of a simply supported rectangular plate is propor-

tional to the square root of the following expression:

Dll 2(D12 + 2D66) D22
fl L + L2W2 + (3.3)
L4 L2W2 W4

where L = 20 in and W = 15 in are the length and width of the plate, the Dij terms

are the coefficients of the bending stiffness matrix (cf. Appendix A):

h 4 Zn t
DI = U, + 2 tk(3 t) cos2xk + U3 tk(3z/ t) cos4xk (3.4)
12 k3 3
k= 1 k= 1

4u2_ 4 Z t
D22 1 U t)tZ cos2xk + U3 tk(3 t) cos4xk (3.5)


D16 = U z U3) cos 4Xk (3.6)

k=l
D12 U4 1 U3 1k k k 1 C Xk 3.7)
k=1

where the U terms are the material constants given in Table 3-1, the ply thickness

tk was fixed at 0.005 in, and zk refers to the position of the kth ply in the laminate,

as illustrated in Figure 3-7.

Like the All maximization problem, this problem is an ADF, therefore UMDA

theoretically converges to the optimum x* 600, i = n. However, contrary to the












ZO




Zk-1

Zk

Zn-1


-- -------------------
61




--k O-


------------------ --------
06


Znpln



Figure 3-7: Ordering convention for a balanced symmetric laminate.


first problem, the z's in the DiD's give a hierarchical structure to the problem because


plies located in outer l-zv.-r have more weight than those located in inner lv,-'ri, as


can be seen in Figure 3-8 (case n = 2). As a result, the distributions corresponding


to the outermost plies are expected to converge faster than those corresponding to


inner plies. The functions All and fi are unimodal, so that SHC will also yield x*


for the two problems.


240.

220-

200.

180.

160.

140-

120
80
60


0 0


40
20

1


Figure 3-8: Fitness landscape of the first natural frequency fl for n = 2. The variable
xl is the orientation of the outer plies, and x2 is the orientation of the core plies.







49

A competitor for UMDA: stochastic hill-climber

SHC searches the space by choosing an initial point x = (x, x2,... x, at ran-

dom and applying random perturbations to it: the algorithm changes the value of

one variable chosen at random to an .,.i ,i:ent value and accepts the new point only

if it improves the fitness function. The procedure is stopped when a fixed number of

function evaluations has been performed.

Theoretical analysis of SHC

The following analysis considers a stochastic hill climber (SHC) operating on

a unimodal function. If the SHC is at a point where k out of the n variables are

correctly set, the expected time before one of the non optimal variable is perturbed is

n/(n- k). The random perturbation can then take the variable closer to the optimum

or not, with probabilities 1/2 (neglecting distortions due to limits on the values). The

expected time for one beneficial step is then 2n/(n k). Let di denote the average

distance between the i-th variables of a random point and the optimum,

C c
di= x -x4: (3.8)
j= 1

where xj is the jth possible value of the ith variable. In the cases presented here,

the value of all the variables at the optimum is the same (x* = 0 for max All and

x' = 450 for the vibration problem), therefore the average distance to the optimum is

the same for all variables i, di = d (but it varies with the problem). For each variable

that is not correctly set, an average of d steps in the right direction is needed to reach

the optimum. By summing up the expected times of each beneficial step, one obtains

the expected time to locate the optimum from a random starting point that has k










optimal variables


Tk -- (3.9)
i=k

Tk can now be averaged over all random starting points, which yields the expected

convergence time of an SHC on a unimodal function,

1 n j T n-1 t
TSHC Z Tk d 1 (3.10)
kO0 k-0 i=k

It can be shown Gamier et al. (1999) that Equation (3.10) yields a convergence time

of order O(n n n) for large n. Estimated (from Eq. (3.10)) and measured convergence

times (average over 50 independent runs) are compared for both the max All and

the vibration problems in Figure 3-9. The qualitative prediction of the dependency

of the number of function evaluations in terms of the dimension n is correct. Because

d is smaller in the vibration problem than in the max An problem, convergence is

faster in the former case, which is also correctly predicted.

6000
calculated, Al 1
measured, All
8 5000 calculated, vibration
S-e- measured, vibration
2)
> 4000

3000-
E

S2000

o 1000-

0
0 50 100 150 200
n

Figure 3-9: Expected convergence time of the stochastic hill-climber for the max A11
and vibration problems. SHC finds the optimum in O(n In n) evaluations.









Theoretical analysis of UMDA


A univariate marginal distribution algorithm is now considered. In Miihlenbein

et al. (1999), the behavior of a UMDA with truncation selection was studied for two

test functions, Onemax and Int, that have common features with the max A1 and

the vibration problems. In the Onemax problem, the number of one's in a binary

string is maximized. As in max All, the function is separable, and each variable has

the same contribution to the objective function. The Int function,


Int = 2-1x, (3.11)
i=1

is also maximized on binary strings. As in the vibration problem, the function is

separable and there is a gradual influence of the variables on the function. In the

vibration problem however, the difference in sensitivity of the objective function to

each variable is lower than in Int. If the population size, /, is larger than a critical

value p*, the authors show that the expected number of generations to convergence

in distribution, denoted by N,, and defined by pNg(x*) = 1, is


Ng, O( n) for Onemax (3.12)

Ng, O(n) for Int (3.13)


The larger number of generations seen on Int is due to the different variable weights

in the objective function. For a truncation rate r = 0.5, the selection in the first

generation is exclusively based on x,, in the second generation, it is based on x,_-1,

etc. The discovery of the optimum is sequential in variable values, whereas some level

of parallelism can be achieved on less hierarchical objective functions.







52

The expected number of objective function evaluations to convergence is


Nf = Nvp* (3.14)


No analytical expression for m* was given in Miihlenbein et al. (1999). An approxi-

mation to p*, p* is now proposed based on the initial random population sampling,

and neglecting variable values lost during selection. The probability that a given

variable value is not represented in the population is ((c l)/c)". The probability

that the values making up the optimum, x*, have at least a sample in the initial

population is

Ppop (1 (3.15)

For a given Ppop (typically close to 1), the critical population size is estimated from

Equation (3.15),

'-* n Pp )) O(ln()) (3.16)
ln(c/(c 1))

From Equations (3.12) to (3.16), the order of magnitude of the number of evaluations

to convergence is


Nf O(v ln(n)) for Onemax and (3.17)

Nf O(nln(n)) for Int (3.18)



Experimental results


The algorithms were applied to the problems for five different numbers of vari-

ables n = 12, 20, 50, 100, and 200. The selection ratio r of the UMDA was kept

constant at 7 = 0.3 (as recommended in Miihlenbein and Mahnig (1999)). Several







53

population sizes A (50, 100, 500 and 1000) were tried in order to obtain an efficient

scheme and allow a fair comparison with SHC.

Two criteria were used to compare the performance of the algorithms: the num-

ber of analyses required to reach I'. reliability (defined as the probability of find-

ing the optimum, estimated over 50 independent runs), and the number of analyses

needed until the average best fitness reaches '-'. of the optimal fitness.

Figure 3-10 presents the number of evaluations to ',-'. of the maximum fitness

against the number of variables for SHC and four different population sizes of UMDA.

Clearly, SHC converges faster than UMDA for all the numbers of variables investi-

gated. The evolution of the cost of SHC is close to linear, which confirms the results

obtained in the previous section and reported in Pelikan et al. (2000) for the Onemax

problem. For UMDA with a given population size, the number Ns98 of evaluations

needed to reach '-'. of the optimal fitness increases sub-linearly: fitting a model

N98 = anflnn to the experimental data yielded N98 = 361 n0371 nn for A = 500

(R = 0.99) and N98 = 650 n038 Inn for A 1000 (R = 0.99), confirming the valid-

ity of the estimate u/ln n as. Larger populations are more expensive, but smaller

population may fail to converge for large n. This is the case when a population of 50

individuals is used to solve the Max All problem and n > 50: the average maximum

fitness never reaches *'-'. of the maximum fitness. This can be explained by the fact

that when smaller populations are used, the chance of losing particular values of the

variables is higher, which prevents the algorithm from finding the optimum, as was

discussed in Section 3.2.2.

The effect of the loss of variable values for small populations is visible in the











x104
SSHC
SUMDA pop = 50
2.5 UMDA pop = 100
UMDA pop = 500
M UMDA pop = 1000
E
S2-

0
S1.5
0)

S 1


0.5


0
0 50 100 150 200
n

Figure 3-10: Number of analyses until the average maximum fitness reaches I-'~. of
the optimal fitness, max All problem. UMDA requires populations of increasing sizes
to avoid premature convergence due to large fitness variance. SHC does not exhibit
such discontinuities in the performance.



reliability: for each problem size n, there exists a minimum population size below


which SI' reliability is never reached because of premature convergence of the dis-


tributions. This minimum population size was m* = 100 for n = 12, m* = 500 for


n = 20, 50 and 100, m* = 1000 for n = 200.


The algorithms were then applied to the vibration problem for the same five


problem sizes. The number of evaluations necessary for the average maximum fitness


function to reach I-'. of the optimal fitness is shown in Figure 3-11. The results are


similar to those obtained for the Max All problem. The cost of SHC is still close


to linear in the number of variables. However, the number of evaluations needed by


the two algorithms to reach i'-'~. of the maximum fitness is smaller than on the Max


All problem. For instance, for n = 12, SHC needs 64 evaluations on the vibration








55

problem, against 160 on the Max All problem. In the case n = 200, it requires 1,402

analyses for the vibration problem, against 2,923 for the Max All problem. Similarly,

the number of ain -il -, needed by UMDA with a population of 1000 individuals

decreases from 4,175 analyses to 2028 analyses for n = 12 and from 26,322 analyses

to 17,531 analyses for n = 200. The faster convergence toward high fitness regions for

the second problem can be explained by the fact that a large part of the response is

governed by outermost plies, so that most of the fitness improvement can be achieved

by determining the value of these influential plies. In addition, the fact that the

optimum angle (600) is close to the center of the domain helps SHC by reducing the

average number of steps it has to take.

Vibration problem
18000
SHC
16000 UMDA pop = 500
UMDA pop = 1000
x 14000
E
S12000

) 10000

S8000

U 6000
0
4000

2000

0 50 100 150 200
n

Figure 3-11: Number of analyses until the average maximum fitness reaches '-'. of
the optimal fitness, vibration problem. When the influence of some variables on the
fitness function is stronger than some others', the effect of dimensionality on UMDA's
effectiveness is reinforced.


If the hierarchical structure of the problem allows a rapid convergence to high

fitness regions, it also causes numerical difficulties for UMDA. The convergence of the








56

probability distribution for the innermost and outermost plies for the case n = 100,

m = 500 are presented in Figure 3-12. On this hierarchical problem, it appears

very clearly that the algorithm proceeds from the outside to the inside, starting

with the more influential variables, and determining the inner variables only at the

very end. This mechanism is responsible for the loss of variable values for the less

average probability distribution for the innermost ply average probability distribution for the outermost ply
1 1


60 60
75 T -75
06 90 06 90

04 04

02 02


1 2 3 4 0 1 2 3 4
number of analyses x 104 number of analyses x 104

(a) Innermost ply (b) Outermost ply

Figure 3-12: Evolution of the probability distributions.



influential inner plies: in the early stages of the search, the selection of good points is

dominated by the outermost variables: the mean of the fitness of dominated variables

is only slightly greater for the optimum value than for non-optimum values, and its

variance is large, as was remarked in Section 2.4. As a result, points which contain

the optimum value of the outermost plies but not of the innermost plies get selected,

potentially leading to the disappearance of these values in the distribution if too small

a population is used. We observed that larger populations have to be used in order

to prevent the loss of values. The minimum population sizes for this problem were

m* = 500 for n = 12 and n = 20, m* = 1000 for n = 50. In the cases n = 100 and

n = 200, the algorithm did not reach S '. reliability for the population sizes tested








57


within the maximum number of 40,000 analyses used in this work.


SHC, however, is unaffected by the problem's hierarchical structure because it


merely compares two neighbor points. Figure 3-13 shows the two performance mea-


sures on the two problems. Both the number of evaluations to '-' of the maximum


fitness and the number of evaluations until II' '. reliability is achieved are lower for


the vibration problem than for the Max All problem: the location of the optimum


in a center area of the design space makes the vibration problem easier for SHC.

6000
Max Al, 80% reliability
Max Al, 98% max. fitness
5000 + Vibration, 80% reliability
Vibration, 98% max. fitness

4000
(U) /
) /
() /
S3000


2000


1000


0
0 50 100 150 200
n

Figure 3-13: Comparison of the performance of SHC on the Max An1 problem and
the vibration problem.



This study reveals that the potential benefits of a probabilistic optimization


algorithm do not lie in their performance in the face of increasing dimensionality: if


it is true that for a given population size, UMDA would .i-i~!,iii ,i ically outperform


SHC, which needs close to linearly increasing numbers of random trials to find the


optimum. To maintain a sufficient accuracy of the statistical model, large populations


have to be used in UMDA, thereby neutralizing the advantage of that model. Like







58

other EAs, EDAs are costly algorithms. Their advantage is observed on problems

that cannot be handled by simpler algorithms, such as multimodal problems, as will

be demonstrated in Section 3.3.

3.3 Investigation of possible modifications to UMDA: memory and
mutation

We have seen in Section 3.2.2 that UMDA variants that had a small population

size and a large selection pressure exhibited the fastest convergence, but that they also

di-!1 .i, 'it a poor reliability because when a value disappeared from the distribution

at a given iteration, no recovery mechanism allowed it to be reintroduced at a later

generation. When the lack of diversity injection mechanism is associated with a very

selective scheme that converges rapidly to a small number of values, the algorithm

becomes strongly exploitative, and premature convergence is very likely.

The reliability of these variants may be improved by three different strategies:

slowing down the convergence of the probabilities using memory, providing an allele

recovery mechanism in the form of mutation, or specifically not allowing probabilities

to vanish. This section investigates the effect of memory, mutation, and of imposing

lower bounds on the probabilities on the performance. The tests were performed on

the 10-variable A11 maximization problem introduced in Section 3.2.1.

3.3.1 Memory

A simple way of preventing premature loss of diversity due to sampling errors is

to use a conservative updating method that makes it possible to adjust the influence

of incoming observations, as proposed by B ilii (1994). The new probability distri-

bution pt+l(xk) is obtained as a linear combination of the old distribution pt(xk) and










the observed frequencies fk:


1
pt+ (np + fk (3.19)


where m is a "memory" parameter: when m = 0, the new probabilities are simply

the marginal frequencies (standard UMDA), when m / 0, the algorithm mitigates

potential sampling errors in the frequencies by giving them a smaller weight. The

value of m reflects the user's confidence in the frequencies' accuracy.


To investigate the effect of memory, we considered two variants of UMDA: Al,

the best scheme found in Section 3.2.2 {p = 50, A = 166} and A2, the algorithm

that di-i'1 -i, -1 the fastest initial convergence { = 10, A 33} (see Figure 3-5). Six

values of m were tested: m = 0, 1, 2, 5, 10, and 20. The reliability of the two variants

for the five values of memory are compared to the basic UMDA in Figure 3-14. Not



1 --- -- ----+--+- ----
/ I
0.9 I
I




In i





0.2 ,' i' ; /^ _.." m"= 2

Figure 314: Effect of memory on UMDA with two parameter settings, max A





problem (n = 10): { = 10, A = 33} (solid lines) and { = 50, A 166} (dashed
lines).
0.2- m= 2
m=5


0 500 1000 1500 2000 2500 3000 3500
number of evaluations

Figure 3 14: Effect of memory on UMDA with two parameter settings, max All
problem (n 10): {/i 10, A 331 (solid lines) and {p 50, A 1661 (dashed
lines).







60

surprisingly, adding memory to the best scheme Al caused the reliability to deterio-

rate: the population sizes had been adjusted for the simple UMDA, but they are no

longer appropriate when substantial modifications are made to the algorithm, hence

the decrease in reliability. In contrast, A2's reliability improves dramatically when

memory is used: the final reliability increases monotonically with m. However, this

improvement is achieved at the cost of convergence velocity, as large memory clearly

adds inertia to the search by giving a large weight to past observations. As a result,

the benefit of memory is debatable, and increasing population sizes may be a more

effective way of preventing distribution degeneracy, as iir--. -1. I by the comparison

between the memoryless Al and the A2 versions, which incorporate memory.


3.3.2 Mutation


A customary strategy for preventing premature convergence in evolutionary algo-

rithms is to use a perturbation operator called mutation. In the context of estimation

of distribution algorithms, two approaches can be adopted to implement mutation:

one can either apply the perturbations to the points or directly to the distribution.

Few instances of mutation are provided in the literature. B iuli (1994) proposed a

mutation consisting in shifting the probabilities by a fixed value with a given prob-

ability. He concluded that while some improvement could be achieved, the operator

was not as critical as in standard genetic algorithms. In this work, we chose the

alternate approach of applying perturbations to the points generated by the (unper-

turbed) probability distribution. Indeed, in EDAs, the statistical model reflects our

knowledge of promising regions, hence there is no reason to degrade this information









arbitrarily. We prefer to provide a separate diversity injection mechanism: if the

proposed perturbation is validated through selection, it will be incorporated into the

model at the next iteration.

The mutation operator used in this work was an I[i i:ent mutation", where

the value of each variable Xk was changed with probability pm to one of the two

neighboring values6 (for instance, 300 could be changed into 150 or 450 with equal

probabilities if the values were chosen from {0, 150, 300, 450, 600, 750, 90}).

UMDA was applied to the 10-variable An maximization problem presented in

the previous sections with seven values of the mutation rate: pm = 0, 0.005, 0.01,

0.02, 0.05, 0.1, 0.2, and 0.3. As in the previous section, two schemes were considered,

Al, the best scheme found in Section 3.2.2 ({p = 10, A = 33}) and A2, the algorithm

that di-i' it .1 the fastest initial convergence ({p = 50, A = 166}). Figure 3-15

shows the influence of the mutation rate on the reliability of the algorithm for the

two variants.

Like memory, mutation had a negative impact on Al, because the values of A and

p chosen to maximize the reliability of UMDA were no longer the best in a modified

context. In contrast, using mutation in combination with A2 led to a dramatic

improvement of the reliability of the algorithm: while the original algorithm had a

very low maximum reliability of 1I'. even with a low mutation rate of 0.005 the




6 The boundary values 0 and 900 required a special treatment: when mutation
was applied to these values, the variable had equal chances of remaining unchanged,
or being shifted to the next value (150 for an initial value of 0 750 for an initial
value of 900).










1-
0.9
0.8-

0.7 -
2 0.6
C
'0.5- p = 0 00
0.4- U U II p,,,- U U1
0.4-
0.3 / Pm m =0.05
pm = 0.05
0.2 t Pm = 0.1
0.1 C Pm= 0.2
pm = 0.3
0 200 400 600 800 1000 1200 1400 1600 1800 2000
number of evaluations

Figure 3-15: Influence of mutation on the reliability for two UMDA variants: the
best scheme without mutation {p = 50, A = 166}, and the one that di-,~l-i the
fastest initial convergence {p = 10, A = 33}. Mutation dramatically improves the
former variant's performance by preventing complete disappearance of the optimal
variable values, but affects the latter's reliability negatively, because it perturbs the
estimation of distributions.


algorithm's reliability reaches I' 1. in approximately the same number of evaluations

as Al. The reliability can be further improved by increasing p,, the best performance

being observed for p, = 0.2. Beyond this value, a sharp decline in the reliability takes

place. The low sensitivity of the reliability to the mutation rate p, over a wide range

of values constitutes a positive attribute, as it reduces the amount of tuning necessary

to use the algorithm.


Unlike memory, mutation did not affect the convergence velocity substantially.

when p, is low, mutation acts as a background operator that compensates for detri-

mental effects of working with a finite population. The role of mutation in the opti-

mization can be observed in Figure 3 16: in the first five iterations, it prevents the

optimal value xz = 0 from disappearing from the distribution; the algorithm then







63

identifies it as the optimal value through the process of selection and distribution

estimation.


1
0.9
0.8
0.7
0.6









1 2 3 4 5 6 7 8 9 10
S0.5
0.4







number of iterations

Figure 3 16: Effect of mutation for pm 0.005. Thanks to mutation, the optimal
value x1 -- 01 0, which would otherwise have disappeared from the distribution, is
reintroduced through mutation. Since the perturbation level is low, the distribution
eventually converges to the optimal value.



3.3.3 Bound on the probabilities


We have seen in previous sections that one of the major concerns in estimation

of distribution algorithms was to maintain the algorithm's ability to visit any point in

the design space. This can be most easily achieved by formally preventing marginal

probabilities to fall below a threshold c. Let Pki p(xk ai). The following procedure

was implemented for each marginal distribution p(xk):

1. if a probability Pki is smaller than c, it is set to c;

2. the sum of the differences C pki of all the corrected probabilities is evenly

distributed over the other values.
distributed over the other values.








64

The effect of imposing a lower bound on the marginal probabilities was investi-

gated for the two UMDA schemes used in previous sections. The reliability for three

values of c is shown in Figure 3-17 (values of c are given as fractions of the uniform

probability pki = /c). As for the other diversity injection mechanisms presented



I ....o ... o, .. 0.o ,. .o .o .o... ... o
0.9 -

0.8

,0.7
2 0.6
C


0.4 -
0.4

3 1 no bound
0.2- se=1/(40c)
-- e=1/(20c)
0.1 es=1/(lOc)
-,e- e=1/(5c)
0 500 1000 1500 2000 2500
number of analyses

Figure 3-17: Effect of imposing a lower bound on the probabilities on the reliability.




previously, imposing bounds on the marginal probability had an adverse effect on the

reliability of the already close to optimal scheme Al. However, it had a dramatic


impact on A2: with values of c between 0 and the reliability reached 10l'.

in less than 2,000 evaluations without appreciable deterioration of the convergence

velocity.


3.3.4 Elitism


A strategy often used to guarantee a monotonic increase of the fitness in evo-

lutionary computation is elitism, which consists in automatically copying the best








65

1 .1 .-" -+ -. .+ .+ .+ ..+ ---+---+-- + +-- +

0.9 -- X=33, p=10, non-elitist
X- X=33, p=10, elitist
0.8 ...... =166, =50, non-elitist
0.7.... =166, pi=50, elitist



0.5-

S0.4
0.3

0.2

0.1
0 iiiff i-^ *-------------
0 500 1000 1500 2000 2500 3000 3500
number of analyses

Figure 3-18: Effect of an elitist strategy on reliability.




parent in the child population. The underlying principle is that the population con-

stitutes the memory of past observations, hence good individuals encountered during

the search should not be lost, as they are evidence for good regions. In the EDA

framework, the notion of individual is less important, as the utility of populations of

points is only to characterize the distribution of promising regions; after information

has been extracted from good points, the population becomes useless. However, some

authors have proposed replacement strategies allowing a portion of a population to be

kept for the next iteration (e.g. Cho and Zhang, 2002; Bosman and Thierens, 2000).

The influence of elitism was investigated for the two UMDA schemes Al and A2: at

each iteration, A 1 were created by sampling from p(x), and the best point of the

previous generation was added to obtain a population of A points. The reliability of

elitist algorithms is compared to that of non-elitist algorithms in Figure 3-18. Elitism

did not help for either of the two UMDA schemes considered: in the case of Al, the







66

population sizes were large enough to guarantee a good accuracy of the distribution,

so that no information about good regions was lost and elitism was not needed; in

the case of A2, copying the best point of a population in the next iteration's popula-

tion reinforced the algorithm's tendency to premature convergence, which caused the

reliability to further deteriorate.


3.3.5 Conclusion of the parameter study


In this section, four possible modifications to the original univariate marginal dis-

tribution algorithm were investigated. The main goal was to provide a mechanism to

prevent premature convergence of the search distribution. Three different approaches

were proposed: memory, mutation, and a lower bound on the marginal probabilities.

All three strategies yielded substantial performance improvement. In particular, they

made the use of small populations viable, thereby allowing savings of function eval-

uations, either by allowing lost values to be recovered, or by preventing the loss of

variable values. The three approaches share the common characteristic that they can

hamper final convergence when used too massively. This calls for adaptive strategies

to decrease the influence of these perturbing operators over time, which is beyond

the scope of this study. Finally, a study of the effect of elitism was carried out for

complete this strategy commonly did not lead to a performance improvement for the

two variants investigated.


3.4 Comparison on three problems


In this section, we compare the performance of UMDA to two other stochastic

optimization algorithms: a stochastic hill-climber, and a standard genetic algorithm.







67

The three algorithms are applied to three laminate optimization problems carefully

chosen to assess the relative performance of the algorithms on representative fitness

landscapes.


3.4.1 Presentation of the algorithms


An UMDA with truncation selection of ratio r = 0.3 was compared to two

stochastic optimization algorithms: a stochastic hill-climber (SHC), presented in Sec-

tion 3.2.3, which is a point-based random search algorithm, and a standard genetic

algorithm (GA) (see Appendix D for an introduction to genetic algorithms) with rank

proportional roulette selection (i.e. linear ranking) selection and two-point crossover

with probability pc = 1.0. No elitist strategy was implemented in UMDA and GA: we

want to compare the fundamental search mechanisms of the algorithms. The study

was primarily focused on UMDA and SHC: a limited parameter study was performed

to obtain good values of the mutation rate pm and population size A. GA's perfor-

mance for the best UMDA setting is provided to allow readers more familiar with

that algorithm to assess UMDA's performance.


3.4.2 Constrained maximization of the first natural frequency


The first problem was maximizing the first natural frequency of a simply sup-

ported graphite-epoxy laminated plate of length L = 50" and width W = 15" subject

to a constraint on the effective Poisson's ratio vi < vf < v", with vi = 0.48 and

v" = 0.52. The first natural frequency is given by

F 72 /Dl 2(D12 + 2D66) D22
F h + L2+ (3.20)
z/0,t L4 L2W2 W4







68

where h is the total laminate thickness, p, designates the mass density, and the Di's

are the bending stiffness coefficients.

The effective Poisson's ratio is given by

A12 U1 2V + U3V3 (3.2
ve=ff (3.21)
A 2 2 4 U3 V3

where the in-plane lamination parameters V* and V* are obtained by


V* cos 20k, V3 1 cos (3.22)
k= k=1

The material properties used for this problem are shown in Table 3-3.
Table 3-3: Material properties of graphite-epoxy.

Longitudinal modulus El 2.18 x 107 psi
Transverse modulus E2 1.38 x 106 psi
Shear modulus G12 1.55 x 105 psi
Poisson's ratio v12 0.26
Weight density p 0.057 lb/in3
Ply thickness t 0.005 in


None of the three algorithms accommodates constraints, therefore a penalty ap-

proach was used, where the fitness function Fp of infeasible designs was decreased in

proportion to the constraint violation:


F F(x) if g(x) < 0 (feasible)
Fp =(3.23)
F(x) pg(x) if g(x) > 0 (infeasible)

where p is the penalty parameter whose value was adjusted empirically to ensure

that the algorithm yielded feasible designs. We used p = 2 x 103 for this study. The

constraint term g(x) was defined as


g(x) max (i Veffi(X) e()(3.24)
g(x) = max 1 1 (3.24)
7,1 Vu







69

The constraint on the Poisson's ratio forces the points to remain in a narrow channel

in the design space, which makes the problem particularly difficult for hill-climbing

algorithms because many of the random perturbations result in infeasible designs.


Clearly, this problem did not satisfy the variable independence assumption, as

the value of one variable influences the optimal value of other variables through the

constraint.


Two numbers of variables were considered: n = 8 and n = 15. Without the con-

straint, the optimal orientation would be 900 for all the plies. The effective Poisson's

ratio would then be eff = v21 = 0.0165. The Poisson's ratio constraint forces 300,

450 600, and 750 plies into the inner lvr-iS of the laminate, where they are the least

damaging to the frequency. The optimum' for n = 8 was [902/ 75/ 455/ 30],,

which has a first natural frequency of F = 670 Hz and an effective Poisson's ratio of

vff 0.482. For n = 15, the optimum was [904/ 75/ 602/ 455/ 3051,, with

F = 1, 262.6 Hz and vff = 0.481.


For n = 8, two population sizes, A = 20 and A = 50, and three mutation rates,

pm = 0.1, pm = 0.2 and pm = 0.3 per variable were tried for UMDA. In order to select

the best scheme, the reliability reached at 2000 evaluations was used as criterion. The

highest reliability (S,' .) was achieved with A = 20 and p = 0.2. A similar parameter

study was conducted for SHC with pm ranging from 0.1 to 0.5, and the highest




7 For this problem, as for all the problems addressed in this work, the best solution
found in all the runs performed is considered as the optimum, unless the global
optimum is known a priori.








70


reliability was achieved for p, = 0.4. Figure 3-19 compares the mean best fitness


and reliability of the three algorithms for these best settings (the same population size


and mutation rate were used for GA and UMDA, to ensure comparable variability


of the search distributions). SHC converges faster than GA and UMDA to high


700 1 ..-~ ---------
.r...._._ 09
650 0 8
07
o 600
S06
S550 o 05
a 04
E 500 03

02
450 --SHC SHC
UMDA 01 UMDA
GA GA
400 01
0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000
number of evaluations number of evaluations

(a) (b)

Figure 3-19: Mean best fitness (a) and reliability (b) for SHC, UMDA, and GA
for the constrained maximization of the first natural frequency of a laminated plate
(n = 8). When the number of variables is small, SHC can progress along the tunnel,
and its performance is comparable to that of UMDA.



fitness regions, as shown in Figure (a), however this advantage does not translate


into a higher reliability than UMDA. SHC's failure to finalize the search can be


attributed to the high mutation rate, which is an indication that an adaptive scheme


may improve the performance. GA's apparent poor reliability is the result of a weaker


selection procedure (linear ranking, cf. Section 2.4). However, considering that it is


one the most commonly used, the observed behavior can arguably be regarded as


representative of a typical GA. Clearly, both SHC and UMDA find the optimum


more reliably than GA on this example.


For n = 15, a population size of A = 100 individuals and a moderate mutation








71


probability pm =0.1 were chosen. The larger population is justified because more


individuals are needed to ensure that all variable values are present in the population.


For SHC, the best mutation probability was found to be pm =0.2 (tested values were


pm = 0.1,0.2,0.3,0.4,0.5). Figure 3-20 compares the performance criteria for the


three algorithms. This time, UMDA seemed to benefit from the use of a global


1400 7 SHC
UMDA
06 -GA
1300
05
S1200
90 :04
1100 -
j 03
E 1000
02

900 -- SHC 01
UMDA
GA
800 0
0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000
number of evaluations number of evaluations

(a) (b)

Figure 3-20: Mean best fitness (a) and reliability (b) for SHC, UMDA, and GA for the
constrained maximization of the first natural frequency of a laminated plate (n = 15).
When the number of variables is sufficiently large, progressing along the "channel"
becomes increasingly difficult for SHC, and UMDA's global approach becomes more
effective.



probabilistic model, which allows it to escape local minima and was able to reliably


find the global optimum: after 10,000 evaluations, the reliability of the optimization


had reached 6 !' whereas the SHC only found the optimum in GI of the runs. In


71' of the runs, SHC converged to a high quality solution but failed to yield the true


optimum. For instance, one of the solutions was [906/ 60/ 4510/ 30],, which


had a fitness of F = 1, 279.3 (F = 1, 257.9, v = 0.481). In order to obtain the global


optimum [904/ 75/ 502/ 455/ 305,1 (Fp 1, 262.6, F = 1, 262.6, v = 0.481),


six variables have to be mutated. However, all single mutations lead to a reduction







72

in the fitness function, either because they make the design infeasible (variable 5)

or because they decrease the vibration frequency. Consequently, multiple mutations

must occur simultaneously for the fitness function to improve. The probability of

that event decreases as n increases, thus making further progress of SHC unlikely.

These results agree with He and Xin (2002) in which an SHC was shown to have an

exponential time complexity for a multimodal function, while the cost of a population

based EA was only polynomial.

3.4.3 Minimize A66

SHC is misled by local minima, while the relationship between UMDA and the

objective function is more complex. It is nevertheless known that, if the objective

function is separable8 and the population size is larger than a critical value A*, UMDA

converges to the global optima (\! 11!i1. :l hein and Mahnig (1999)). An example of such

an objective function where the reliability of UMDA tends to 1 while that of SHC is

nearly 0 is the minimization of the in-plane shear stiffness of a composite laminate,

A66, over ply orientations that are bounded between 0 and 75,


min A66 (3.25)
0 o i < 75


where

A66 -Uh 3 ti cos 4x (3.26)
i=




8 A more general result is given in Milhlenbein and Mahnig (1999) where the con-
vergence of 1-" '. I, d Decomposition Alsuiihmi (FDA) to the optima is proved for
additively decomposed functions. Separable functions are a special case of additively
decomposed functions and UMDA is the corresponding simplified FDA.







73

The global optimum is x* = 00, i = n. Replacing any of the x* with 750 creates

a local optimum whose basin of attraction starts at x = 45. For an n-dimensional

case, there are 2" local optima. Numerical experiments with n = 12 averaged over

50 runs confirm that the SHC reliability is 0 (it is theoretically (45/75)12 = 2.10-3)

while that of an UMDA with m = 500 reaches 1 after 3,500 analyses. Figure 3-21

shows the evolution of the probability distribution of the outermost ply p(xi). It

is interesting to note that in the early stages of the search, both the probability of

750 and the probability of 0 (the two local optima) increase. But after about 2,000

analyses, the probability of 750 starts to decrease and the algorithm converges to the

global optimum.









0.8 37.5
50
0.7 62.5
75
0.6

0.5

0.4

0.3

0.2

0.1

0
0 2000 4000 6000 8000 10000
number of analyses

Figure 3-21: Evolution of the marginal distribution p(xl), UMDA, max A66 problem.









3.4.4 Strength maximization

Many practical laminate optimization problems exhibit a multimodal relation-

ship between the variables and the fitness function. A typical example of such behav-

ior is the maximization of the strength of a laminate. In this section, we considered

the problem of maximizing the load factor As, using the first-ply-failure criterion

based on the maximum strain (Giirdal et al., 1998, C'!I pter 6), for a glass-epoxy

laminate subjected to the in-plane loading N1 -1000 103 N/m, Ny = 200 103 N/m,

N1y = 400 103 N/m:

ult ult nult
1 2 712
maximize A = min mmin
k=1 1(k) 2(k) 712(k))

where the load factor As is the coefficient by which the load has to be multiplied

for the structure to fail. The material properties used for this problem are shown in

Table 3-4. The total thickness of the laminate was h = 0.02 m.

Table 3-4: Material properties of glass-epoxy.

Longitudinal modulus El 69.0 GPa
Transverse modulus E2 10.0 GPa
Shear modulus G12 4.5 GPa
Poisson's ratio v12 0.31
Material invariant U/ 31.58 GPa
Material invariant U2 27.91 GPa
Material invariant U3 6.48 GPa
Material invariant U4 9.63 GPa
Material invariant U5 10.98 GPa
Tensile strength Xt 500.0 MPa
Compressive strength Xc 410.0 MPa
Tensile strength Yt 35.0 MPa
Compressive strength 110.0 MPa
Shear strength S 70 MPa


We considered the case n = 8 design variables. The laminate is subjected to an












in-plane loading: N,


1000 kN/m, N = 200 kN/m, Ny = 100 kN/m. For this


problem, the optimum laminate was [06/90101] or its permutations, and the optimum


load factor was A8


5.39. Depending on the orientation of the fibers, one of the


three possible failure modes (fiber failure, matrix cracking, shear failure) becomes


critical. The combination of these three failure modes results in a multimodal fitness


function9


55

5-

45-

4- /

E 35

3

25
0 200


07
06
S05
04
03


400 600
number of evaluations


-- SHC
---UMDA
...... GA
800 1000


200


400 600
number of evaluations


Figure 3-22: Reliability (a) and mean best fitness (b) of UMDA and SHC for the
strength problem. Unlike SHC, which was trapped in local optima, UMDA was able
to find the global optimum reliably. When the maxima of the objective function are
far apart in the Hamming space, SHC is not able to escape local maxima, hence its
poor reliability. In contrast, UMDA's global search strategy allows it to locate high
function evaluation regions and to reliably find the global maximum.


For UMDA the population size was A


50 and a moderate mutation rate of


p, =0.1 per variable was implemented to prevent premature convergence. Several


variants of SHC were compared to obtain the best competitor to UMDA. For this






9 Note that its is not necessary to consider all the failure modes for the response
to be multimodal, as individual failure modes can be multimodal for specific combi-
nations of material properties and loading.


- SHC
---UMDA
. GA
800 1000


I
'=



...,

,'


. .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..







76

problem, the same .,l.i :ent mutation operator was implemented and the best rate was

pm =0.2 (the same values of A and p, were used for GA). Both the reliability (Figure

3-22-a) and the mean best fitness (Figure 3-22-b) clearly show the superiority of

UMDA for this problem. The reliability of SHC increases faster than that of UMDA,

but culminates at :.' while UMDA was able to converge reliably to the optimum.

SHC often converged to local optima ([08/9081], A, 8 4.97, [06/ 30/908s], A8 = 4.47

or [08/ 30/9061, A = 4.02) and was not able to escape even when large mutation

rates were used because unlikely coordinated mutations were needed to reach the

basin of attraction of the global optimum.

3.5 Conclusion

Univariate marginal distribution algorithms constitute the simplest form of esti-

mation of distribution algorithms. They estimate distributions based on the assump-

tion that the variables are statistically independent in the populations of selected

points (but they are often applied to problems that violate this assumption). In this

chapter, the influence of 1! i' .1, components of the algorithm was investigated on a

laminate optimization problem. The selection pressure determines the convergence

velocity, while the population size affects the variance of the marginal frequencies,

hence the accuracy of the search distribution. UMDA's .i-vmptotic behavior was com-

pared to that of a hill-climbing algorithm. The study concluded that the advantage

of UMDA does not lie in pure convergence velocity. Instead, its global search capa-

bilities become beneficial for functions that exhibit narrow channels, or multimodal

landscapes, as demonstrated on three laminate optimization example problems.















CHAPTER 4
THE DOUBLE-DISTRIBUTION OPTIMIZATION ALGORITHM

In this chapter, we propose a physics-based method for incorporating variable

dependencies into the probabilistic model of promising regions. The apparent com-

plexity of the distribution of selected points can often be explained as the joint action

of a small number of latent variables. The distribution of selected points can be re-

constructed by the cooperation of simple models of the primary variables and of the

latent variables, used as auxiliary variables. This representation of the distribution

of promising points is the basis of the Double-Distribution Optimization Algorithm,

introduced here.

4.1 Motivation

In C'!i lter 3, the distribution of selected points was approximated by a product

of univariate marginal distributions, which neglected any statistical variable depen-

dencies. While this representation yielded comparable to superior performance than a

standard evolutionary algorithm, it may not be appropriate for problems with strong

variable interactions. Figure 4-1 illustrates the impact of the choice of a univari-

ate model when the distribution generated by selection is more complex. In the 2D

problem considered (which is the problem of maximizing the first natural frequency

subject to a constraint on Poisson's ratio introduced in C'!i pter 3), the contours of

the objective function assume roughly the shape of a narrow ridge. After uniform

sampling in the (01, 02)-plane, and application of truncation selection, the promising

77










90
80
70+ +
70 +4 +


0 10 20 30 40'50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
6 4
1 +1 +

00
0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90

(a) (b)

Figure 4-1: Selected points (a) and univariate distribution (b) for (01, 02)
{0, 5,10,..., 85,90}2 for the constrained vibration problem of C'! plter 3. When
the distribution of selected points is approximated by a product of the univariate
marginal distributions p(01) and p(02), high-probability regions do not coincide with
high-fitness regions.




points are distributed as shown in Figure 4-1-a. Ideally, one would want to sample

from the high evaluation regions marked by the selected points. However, when the

joint distribution p(01, 02) is approximated by a product of the univariate marginal

distributions p(01) and p(02), new points will be created according to the distribu-

tion whose histogram is shown in Figure 4-1-b. Clearly, the high probability areas

defined by that distribution do not coincide with high evaluation regions,1 so that

some knowledge about optimal regions contained in the selected points will be lost




1 In a univariate model, maximum probability areas are parallel to the axes.
For example, for x = (xI, x2), the maximum probability at a given value of x2 is
maxIp(xI)p(X2) = p(2) maxip(xi) p(x2)p(xC), where p(x,) is the maximum of
the marginal distribution p(xi): the value of x, that maximizes p(xi, x2) does not
depend on x2.









for the next EDA generation.

One way of improving the distribution accuracy would be to use a higher-order

statistical model that incorporates the conditional probabilities p(021O1). In higher

dimensions, this approach could be generalized by using a B li- i i network to rep-

resent all conditional probabilities p(0ilTr), where 7r designates the parents of the

variable Oi, as proposed by Pelikan et al. (1999) in the B li, -i 'i Optimization Algo-

rithm. However, this approach presents three strong limitations:

the number of parameters required to describe high-order dependencies in the

distribution increases exponentially with the number of variables n (c" 1

parameters are necessary to express the full joint distribution when each of the

n variables can take c values). The computational cost associated with the

estimation of those parameters may become prohibitive as n becomes large;

more importantly, the standard error associated with the parameters increases

with the flexibility of the model. While a flexible model can .-i-,iii11l ically

approximate complex distributions (a fortiori simple distributions), obtaining

an accurate estimation of the parameters may require a very large sample (cf.

Section 2.5.3). In the context of EDAs, this implies that large population may

have to be used, so that larger progress may be accomplished at each iteration,

but at the cost of many function evaluations;

lastly, using a too high-order model violates the premises of optimization, which

is based on the assumption that there exists an underlying structure present

in the distribution p(x): the goal is to infer and use this structure to find the

optima. This implies a trade-off between accuracy and generalization from the







80

sample: high-order models will provide an accurate estimation on the i i ilg

set", but extrapolation to other regions of the design space will be poor.

Alternative approaches are used routinely by statisticians in the field of ex-

ploratory data analysis. The idea is to simplify the data by looking for structure

in the distributions, as will be presented in the next section.

4.2 Principles

4.2.1 Identification of dependencies and data simplification

A typical example of an effort to distinguish between data structure and ran-

dom variations is exploratory factor analysis (EFA), where the goal is to identify

underlying (latent) factors that determine the distribution of a population, and use

them to describe the distribution, thereby as shown in Figure 4-2. By supposing that


V, V2 .......... V_





XX, XX X3X


Figure 4-2: Interpretation of variable dependencies as the joint action of hidden vari-
ables V1,..., Vm. Even if the hidden variables have a simple probability distribution,
the distribution of the observed variables (the X's) can be complex.



the (possibly complex) observed distribution is the result of joint actions of a small

number m of latent variables, whose distribution is simple, one is able to simulta-

neously reduce the number of meaningful variables, as well as the complexity of the

distribution (the low dimensional distribution of independent latent variables are eas-

ier to estimate than the joint distribution of the observed variables). Such methods







81

were implemented by Shin et al. (2001), using several latent variable models: they

modeled the distribution of selected points by Helmolz machines, and probabilistic

principal component analysis (PPCA) to represent apparently complex distributions

as the result of the joint action of several latent variables, whose distribution is simple

(multivariate normal).

This general method is appropriate when nothing is known about the structure

of the distribution, however it only allows simple (typically linear) relationships to

be identified, and learning even these relationships has a computational cost. In

many situations, a priori knowledge about the problem is available. The benefits of

providing the distribution structure was investigated by B lti (2002), who showed

that it led to drastic performance improvement. In this work, we propose a fully

different approach for incorporating information about the structure to enhance the

accuracy of the estimated distribution by directly providing the algorithm with its

structure.

In many instances, variable dependencies among selected points often reflect the

fact that the overall response of the system is really a function of integral quantities,

so that many combinations of the design variables can produce the same response.

For example:

the dimensions of the section of a beam determine its flexural behavior through

the moment of inertia I,

the aerodynamic properties of a vehicle are captured by the drag coefficient CD,

the flow through a porous medium is described by the permeability k,

etc.