lecture08

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

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 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...
View Full Document

## This note was uploaded on 05/06/2010 for the course MSE 211 at Stanford.

### Page1 / 25

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online