lecture12

lecture12 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #12 1 NonlinearOptimizationAlgorithmsIII YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #12 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 #12 3 BarrierFunction ( BLCCP ) minimize f ( x )- μ ∑ n j =1 log x j subjectto A x = b , whereparameter μ> isgiven.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 #12 4 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 #12 5 y a The objective hyperplanes Figure1:Thecentralpathina(dual)feasibleregion. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #12 6 TheAnalyticCenter Inparticular,when μ = ∞ ,themaximizeofthe barrier functiononly: maximize ∑ n j =1 log x j subjectto A x = b , iscalledthe analyticcenter ofthefeasibleset. Theanalyticcenterforthefeasiblesetgivenasinequalities { y : A T y ≤ c } : maximize n X j =1 log s j , subjectto A T y + s = c , y free . maximize log( x 1 )+log( x 2 )+log( x 3 ) subjectto x 1 + x 2 + x 3 =1 Theanalyticcenteris x 1 = x 2 = x 3 =1 / 3 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #12 7 TheBarrierFunctionMethod Wehavea“good” approximate solutionfor ( BLCCP ) minimize f ( x )- μ ∑ n j =1 log...
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lecture12 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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