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lecture11

# lecture11 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 1 NonlinearOptimizationAlgorithmsII YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 2 NonlinearOptimization:PrimalMethods ( NLP ) minimize f ( x ) subjectto g i ( x ) ≥ i =1 ,...,m. PrimalMethodsaremethodsthatworkontheoriginalproblemdirectlyby searchingthroughthe feasibleregionoftheproblem . Forsimplicity,assumethattheconstrainsarealllinear g i ( x ):= a i x- b i . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 3 FeasibleDirectionMethods Searchalonga feasible directionthatisalso descent atthesametime. Lettheactiveconstraintsetbe A ( x k ):= { i : a i x k- b i =0 } ata feasible iterativepoint x k .Thenwesolve: ( FDM ) minimize ∇ f ( x k ) d subjectto a i d ≥ , ∀ i ∈A ( x k ) k d k≤ 1 . Supposetheminimizeris d k .Iftheminimalvalueis ,then x k isalreadya local minimizer ;otherwise x k +1 = x k + α k d k where α k isthe stepsize tomake x k +1 stay feasible and reduce theobjective functionthemost. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 4 ActiveSetMethods Guesswhichconstraintswouldbe active attheminimizer,sayinaworkingset W k .Thenwesolve: ( ASM ) minimize f ( x ) subjectto a i x- b i =0 , ∀ i ∈ W k . TheminimizercanbecomputedfromtheKKTconditionsfor equality constrained problem(whichisgenerallyeasierthan inequality problemtosolve),andletitbe x k . If a i x k- b i < forsome i 6∈ W k ,then ADD thisviolatedconstraintinto W k andresolve(ASM);elseifoneoftheLagrangemultipliers λ i isnegative,then DROP theassociateconstraintfrom W k andresolve(ASM). Ifthereisnoneedto ADDorDROP , x k isa KKTpoint fortheoriginalinequality constrainedproblem. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 5 OtherPrimalMethods Thegradientprojectionmethod :projectthegradientontothefeasibledirection space. Thereducedgradientmethod :likethesimplexmethodbychangingnon-basic variables. Theellipsoidmethod :Thebasicideasofthe ellipsoidmethod stemfromresearch doneinthenineteensixtiesandseventiesmainlyintheSovietUnion(asitwas thencalled).Theideainanutshellistoenclosetheregionofinterestineach memberofasequenceofellipsoidswhosesizeisdecreasing,resemblingthe bisection method. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 6 LinearFeasibilityProblem Themethoddiscussedhereisreallyaimedatfindinganelementofa solutionset X givenbyasystemoflinearinequalities. X = { x ∈ R n : a i x- b i ≥ ,i =1 ,...m } Findinganelementof X canbethoughtofasbeing equivalent tosolvinga linear programming problem,thoughthisrequiresabitofdiscussion. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 7 Twotechnicalassumptions (A1) X iscontainedbya ball centeredatthe origin witha radius R> . (A2) X hasa volume atleast ² n vol S ( , 1) . Here, vol S ( , 1) isthevolumeofthe unitball inthe n-dimensionalspace....
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lecture11 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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