further topics in optimization

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1、ChapterFurtherTopicsinOptimizationThischapterdealt!withtwomajortopics.TheDMisnonlinearprogramming,whichextendsthetechniquesofconstrainedoptimizationofChap.12byallowinginequalitycon•straintsintotheproblem.InCh'lp.12,theconstraintsmustbesati8fk:dasstrictequaIitie~;

2、i.e.,theconstraintsarealwaysbinding.Nowweshallconsiderconstraintsthatmaynotbebindinginthesolution;i.e.,theymaybesatisfiedasinequalitiesinthesolution.Inthesec(lndpartofthischapter,werevertbacktothere

3、ouchedinthepreviouschapters,Theseincludetheindirectobjectivetimcrion,theenvelopetheorem,andthecom.:cptofduality.13.1NonlinearProgrammingandKuhn-TuckerConditionsInthehistoryofmethodologicaldevelopment,thefirstattemptsatdealingwithinequalityconstraintswereconcentra

4、tedonlinearonesonly.Withlinearityprevailinginthecon•straintsasvellasintheobjectivefunction,theresultingmethodologyisquitenaturallychristenedlinearprogramming.Despitethelimitationoflinearity,hllwcvcr,wecouldforthetirsttime,explicitlyspecifythechoicevariablestoben

5、onnegative)asisappropriateinmosteconomicanalysis.Thisrepresentsasignificantadvance.Nonlinearprogramming,alaterdevelopment,makesitpossibleeventohandlenonlinearinequalityconstraint:-.andnonlinearobjectivefunction.Thusitoccupiesamostimport,mtplaceinoptimiL,ationmeth

6、odology,Intheclassicaloptimir.ationproblem,withnoexplicitrestrictionsonthesignsoflhechoicevariables,andy,,'ithnoinequalitiesintheC(ITISlraints,thefirst~orderconditionforarelativeorlocalextremumissimplythatthe:firstpartialderivlitivesofthe(smo(lfh)Lagrangianfuncti

7、onwithrespcettoallthechoicevariablesandtheLagrangemultipliersbezero,Innonlinearprogramming,thereexistsasimilartypeoffirst-ordercondiLio!!.knownastheKuhn-Tuckerconditions.tAsweshallsee,however,whil~theclassicalfirst•orderconditionisalwaysnecessary,theKuhn-TuckereO

8、lldition~cannotbeaccordedthetH.W,KuhnandA.WTucker,"NonlinearProgramming,"inJ.NeyrnJn(ed.),ProceedingsoftheSecondBerkeleySymposium011MathematicalStatisticsandPr