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lecture08

# lecture08 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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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 δ ( ¯ x ) ,anopen ballcenteredat ¯ x havingsuitablysmallradius δ> . 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 ceratinregions. 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 )= y a- y e , ( y a ,y e ) ≥ ,y a ( x- a )=0 ,y e ( e- x )=0 fortwonon-negativenumbers y a and y e . Thesearecalledthe first-ordernecessaryconditions .Here y a and y e arecalled Lagrangeordualmultipliers forthetwoconstraints x ≥ a and x ≤ e , respectively;andthelasttwoequationsarecalledthe complementarity conditions . 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. Theseconditionsarenot,ingeneral,sufficient.Itdoesnotdistinguishbetween localminimizers,localmaximizers,orpointsofinflection.However,ifinadditionto thefirst-ordercondition,thesecond-ordercondition...
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lecture08 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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