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Lecture04CopyRightP.B.Luh1ReadingAssignment:BertsekasSections3.1and3.2LastTime:ConjugateDirectionMethods2. OPTIMIZATIONOVERACONVEXSETNecessaryandSufficientConditionsforOptimalityFeasibleDirectionsandtheGradientProjectionMethods Today:LagrangeMultiplierTheory:NecessaryConditionsforEqualityConstraintsTheLagrangianRelaxationApproachSufficientConditionsandSensitivityAnalysisNextTime:BertsekasSections3.3,3.4,4.3(?),and5.1ModelingOptimizationofComplexNetworkedSystems:ApplicationstoOperationsSchedulingSupplyNetworkManagement
Lecture04CopyRightP.B.Luh2ConjugateDirectionMethodsMakingthebestuseofquadraticpropertiesxk+1=xk+?kdk,?kobtainedbylineminimizationdk=-?f(xk)+?kdk-1,withConditionsforConstrainedOptimality?f(x*)(x-x*)?0?x?X?Anyfeasibledirectionleadstoincreaseinf(x)atleastlocallyFeasibleDirectionsandtheGradientProjectionMethodsConstraintsarealwayssatisfied~thePrimalApproachIfweareontheboundary,projectnegativegradientbackontothesetofactiveconstraintsWith?kchosenbytheArmijoruleorthelimitedminimizationrule,everylimitpointof{xk}isstationaryConvergencesimilartothegradientmethod
Lecture04CopyRightP.B.Luh3LagrangeMultiplierTheory:NecessaryConditionsforEqualityConstraintsA“dual”approachwithoutrequiringtheconstraintstobesatisfiedacrosstheiterationsWeshallfirstpresentintuitiveideasaboutnecessaryconditions,andfollowupbymorerigorousderivationsAmotivatingexampleMinxf(x),withf(x)?x12+x22,subjecttox1+x2=1Whatistheproblem?x1+x2=1Howmanymethodsaretheretosolvetheproblem?Howdotheywork?Prosandcons?Whatisthesolution?
Lecture04CopyRightP.B.Luh4Method1:GraphicalinspectionMethod2:Directsubstitutionx2=1-x1f(x1,(1-x1))=x12+(1-x1)2=2x12-2x1+1?F(x1)?AnunconstrainedoptimizationdF(x1)/dx1=0=4x1
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