lecture12 - Yinyu Ye, MS&E, Stanford...

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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 Nonlinearlyconstrainedproblem ( NCOP ) minimize f ( x ) subjectto A x = b c ( x ) . Weassumethat A hasfullrankand f, c aredifferentiablefunctions,where c ( x )=( c 1 ( x ); ... ; c l ( x )) . TheKKTconditions: C ( x ) s = A x = b- A T y + f ( x ) T- c ( x ) T s = ( c ( x ) , s ) , Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #12 3 where C ( x )= diag ( c ( x )) . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #12 4 BarrierFunctionMethod ( BNCOP ) minimize f ( x )- l i =1 log c ( x ) i subjectto A x = b . TheKKTconditionofthebarrieredproblemis f ( x ) T- c ( x ) T C ( x )- 1 e- A T y = A x = b c ( x ) > , orusing s = C ( x )- 1 e wehave C ( x ) s = e A x = b- A T y + f ( x ) T- c ( x ) T s = . Yinyu Ye, MS&E, Stanford...
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lecture12 - Yinyu Ye, MS&E, Stanford...

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