lecture08

lecture08 - Yinyu Ye, MS&E, Stanford...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #08 1 OptimalityConditionsforNonlinearOptimization YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #08 2 MoreGeneralOptimizationProblems Lettheproblemhavethegeneralmathematicalprogramming(MP)form ( P ) minimize f ( x ) subjectto x F . Inallformsofmathematicalprogramming,a feasiblesolution ofagivenproblem isavectorthatsatisfiestheconstraintsoftheproblem,thatis,in F . Thequestion:Howdoesonerecognizeorcertifyanoptimalsolutiontoa generallyconstrainedandobjectived optimizationproblem? Answer: OptimalityConditionTheory . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #08 3 GlobalandLocalOptimizers A globalminimizer for(P)isavector x suchthat x F and f ( x ) f ( x ) x F . Unlikelinearprogramming,sometimesonehastosettlefora localminimizer ,that is,avector x suchthat x F and f ( x ) f ( x ) x F N ( x ) where N ( x ) iscalleda neighborhood of x .Typically, N ( x )= B ( x ) ,anopen ballcenteredat x havingsuitablysmallradius > . Thevalueoftheobjectivefunction f ataglobalminimizeroralocalminimizeris alsoofinterest.Wecall f ( x ) the globalminimumvalue anda localminimum value ,respectively. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #08 4 ContinuouslyDifferentiableFunctions Theobjectiveandconstraintareoftenspecifiedbyfunctionsthatare continuously differentiable orin C 1 overcertainregions. Sometimesthefunctionsare twicecontinuouslydifferentiable orin C 2 over certainregions. Thetheorydistinguishesthesetwocasesanddevelops first-orderoptimality conditions and second-orderoptimalityconditions . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #08 5 MotivationfromOne-VariableProblem Consideradifferentiablefunction f ofonevariabledefinedonaninterval [ a,e ] . Ifaninterior-point x isalocal/globalminimizer,then f ( x )=0; ifthe left-end-point a isalocalminimizer,then f ( a ) 0; iftheright-end-point e isa localminimizer,then f ( e ) . Tosummarize:if x [ a,e ] isalocalminimizer,itmustbetrue f ( x )= r a- r e , ( r a ,r e ) ,r a ( x- a )=0 ,r e ( e- x )=0 fortwonon-negativenumbers r a and r e . Thesearecalledthe first-ordernecessaryconditions sincethefirstorder derivativeisusedtocharaterizethecondition,where r a and r e arecalled Lagrangeordualmultipliers forthetwoinequalityconstraints x a and x e , respectively;andthelasttwoequationsarethe complementarityconditions . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #08 6 x a b c e d f Figure1:Globalandlocalminimizersofone-varialbefunction Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #08 7 MotivationfromOne-VariableProblemcontinued If f ( x )=0 ,then f 00 ( x ) isalsonecessary,whichiscalledthe second-ordernecessarycondition ,sincethesecondorderderivativeisused. Theseconditionsarealsosufficientif f ( x ) isaconvexfunction.Buttheyarenot sufficientingeneral;forexample...
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lecture08 - Yinyu Ye, MS&E, Stanford...

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