MMA

Themethodofmovingasymptotes——AnewmethodforstructuraloptimizationWhyisMMAproposed?AmethodforstructuraloptimizationshouldbeflexibleandgeneralItshouldtakeintoconsiderationthecharacteristicsoftheproblemItshouldbestableandgenerateasequenceofimprovedfeasiblesolutionsGeneraldescriptionoftheproblemAwellestablishedgeneralapproachforsolvingtheproblem1.Chooseastartingpoint,andlettheiterationindexk=02.Givenaniterativepoint,calculateandthegradientsfori=0,1,…,m3.Generateasubproblembyreplacing,inP,the(usuallyimplicit)functionsbyapproximatingexplicitfunctions,basedonthecalculationsfromstep24.Solveandlettheoptimalsolutionofthissubproblembethenextiterationpoint,letk=k+1andgoontostep1Theprocessisinterruptedwhensomeconvergencecriteriaarefulfilled,orsimplywhentheuserissatisfiedwiththecurrentsolutionExplanationHowthefunctionsshouldbedefinedGiventheiterativepoint,valuesoftheparametersandarechosen,suchthatThen,foreachi=0,1…m,isdefinedbyWhereallderivativesareevaluatedatWhereAsiseasilychecked,isafirstorderapproximationofat,i.e.Further,thesecondderivativesof,atanypointsuchthat,aregivenby:andThus,sinceisaconvexfunction.Inparticular,atThus,thecloserandarechosento,themorecurvatureisgiventotheapproximationoftheoriginalproblemHeretheparametersare“movelimits”,notverycrucialbuttoavoidthepossibilityofanyunexpected“divisionbyzero”,forexample,Evenifthesimplicityof“fixedasymptotes”isappealing,inordertofullyexploittheflexibilityofMMA,onecanmoveitinsomecleverwaybetweentoiterationsAgeneral,althoughheuristic,ruleforhowtochangethevaluesofisthefollowing:(a)Iftheprocesstendstooscillate,thenitneedstobestabilized.Thisstabilizationmaybeaccomplishedbymovingtheasymptotesclosertothecurrentiterationpoint(b)If,instead,theprocessismonotoneandslow,itneedstobe“relaxed”.ThismaybeaccomplishedbymovingtheasymptotesawayfromthecurrentiterationpointAsimpleimplementationofthis“rule”:sisagivenrealnumberlessthanunity,say,s=0.7,(1)Fork=0and1,let(2)Fork=2,Ifthesignsofareopposite,letIfthesignsareequal,letWemayrefuseto“relax”theasymptotesunlessallthreeofAdualmethodtosolvethesubproblemThelagrangianfunctioncorrespondingtoisgivenbywhichaftertrivialcalculation,equals:WhereAndNext,the“dualobjectivefunction”Wisdefinedasfollows:WhereArtificialvariablesItmayhappen,inparticularduringthefirstiterationsifthestartingpointisbadlychosen,thatasubproblembecomesinfeasibleHowtoobtainareasonablenextiterationpoint,ismodifiedbytheintroductionof“artificialvariables”,i=1,2….m,sowehaveMinimizeS,tAndEachshouldbea“relativelylarge”fixedrealnumber.Obviouslytherearealwaysfeasiblesolutiontothisproblem.Ifthesubproblemisfeasible,andthecoefficientsaresufficientlylarge,thenalltheartificialvariableswillautomaticallybecomezerointheoptimalsolutionofIfthesubproblemisinfeasible,thensomeofthewillbestrictlypositiveintheoptimalsolutionof.However,becauseofthehigh“cost”ofthesevariables,theywillnotbegreaterthanabsolutelynecessary.Thus,thecorrespondingX-solutionisinsomesenseascloseaspossibletobeingfeasibleNumericaltestresults(omit)TheobtainedresultsclearlyilluminatetheimportanceoftheflexibilityofMMA,whichgivestheusersomecontroloftheconvergencepropertiesoftheoveralloptimizationprocess.Conclusion