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 LinearlyConstrainedConvexProblem ( LCCP ) minimize f ( x ) subjectto A x = b x ≥ . Weassumethat A hasfullrankand f isa differentiableconvex function. The KKTconditions : X s = A x = b- A T y + ∇ f ( x ) T- s = ( x , s ) ≥ . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 3 BarrierFunction ( BLCCP ) minimize f ( x )- μ ∑ n j =1 log x j subjectto A x = b . TheKKTconditionofthe barrieredproblem is ∇ f ( x ) T- μX- 1 e- A T y = A x = b x > , or X s = μ e A x = b- A T y + ∇ f ( x ) T- s = ( x , s ) > . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 4 TheBarrierFunctionMethod Wehavea“good” approximate solutionfor ( BLCCP ) minimize f ( x )- μ ∑ n j =1 log x j subjectto A x = b . Then,usingNewton’sstepweupdateittoa“good” approximate solutionto ( BLCCP+ ) minimize f ( x )- μ + ∑ n j =1 log x j subjectto A x = b , where <μ + = γμ<μ, where <γ< 1 iscalledthe reductionratio . Andtheprocessrepeats. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 5 InteriorCentralPathsofLCCP Let ( x ( μ ) > , y ( μ ) , s ( μ ) > 0) bethe(unique)minimizerofthe barriered problem .Then,the centralpath ofLCCPcanbeexpressedas C = { ( x ( μ ) > , y ( μ ) , s ( μ ) > 0):0 <μ< ∞} . Theorem1 LetLCCPhave interiorfeasiblepoints andthebarrierfunctionis bounded forthegivendataset ( A, b ,f ) .Thenforany <μ< ∞ ,thecentral pathpoint ( x ( μ ) , y ( μ ) , s ( μ )) existsandisunique. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 6 y a The objective hyperplanes Figure1:Thecentralpathina(dual)feasibleregion. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 7 CentralPathProperties Theorem2 Let ( x ( μ ) , y ( μ ) , s ( μ )) beonthe centralpath . i) The centralpath point ( x ( μ ) , y ( μ ) , s ( μ )) is bounded for <μ ≤ μ and anygiven <μ < ∞ . ii) For <μ <μ , f...
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lecture11 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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