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1、SIAMREV.c1998SocietyforIndustrialandAppliedMathematicsVol.40,No.2,pp.228{264,June1998003OPTIMIZATIONPROBLEMSWITHPERTURBATIONS:AGUIDEDTOURJ.FREDERICBONNANSyANDALEXANDERSHAPIROzAbstract.Thispaperpresentsanoverviewofsomerecent,andsigni cant,progressinthetheoryofoptimi
2、zationproblemswithperturbations.Weputtheemphasisonmethodsbasedonupperandlowerestimatesoftheobjectivefunctionoftheperturbedproblems.ThesemethodsallowonetocomputeexpansionsoftheoptimalvaluefunctionandapproximateoptimalsolutionsinsituationswherethesetofLagrangemultiplier
3、sisnotasingleton,maybeunbounded,orisevenempty.Wegiverathercompleteresultsfornonlinearprogrammingproblemsanddescribesomeextensionsofthemethodtomoregeneralproblems.Weillustratetheresultsbycomputingtheequilibriumpositionofachainthatisalmostverticalorhorizontal.Keywords.s
4、ensitivityanalysis,parameterizedoptimization,directionaldi erentiability,quan-titativestability,duality,expansionofoptimalsolutions,semi-in niteprogramming,semide niteprogramming,second-orderoptimalityconditionsAMSsubjectclassi cations.49K40,90C31,26B25,41A58,65K05,49
5、M45PII.S00361445963026441.Introduction.Inthispaperwepresentasurveyofsomeresultsonstabilityandsensitivityanalysisofoptimizationproblemswhicharesubjecttoperturbations.Weconsiderproblemsoftheform(1.1)Minf(x;u)subjecttox2(u):(Pu)x2XInmostexamplesconsideredinthispaper,Xis
6、aBanachspace,whilethepertur-bationparameterucanbeascalar,a nite-dimensionalvector,orevenanelementofanappropriatenormedormetricspaceU.Weinvestigatecontinuityanddi eren-tiabilitypropertiesoftheoptimalvaluev(u)andthesetS(u)ofoptimalsolutionsof(Pu)consideredasfunctionsoft
7、heparameteru.Inordertoproceedinouranalysisweneedaconstructivemethodfordescribingthefeasibleset(u).Wesaythatthefeasiblesetisde nedbyabstractconstraintsifitisgiveninthefollowingform:(1.2)(u)=fx:G(x;u)2Kg;whereKisaclosedconvexsubsetofaBanachspaceYandG:XU!Y.Thisinclude
8、sthesituationswhereKisaclosedconvexcone,inwhichcasewespeakofconeconstraints.Animportantparticularcaseiswhenthefeasiblesetis
