lecture15 - Yinyu Ye MS&E Stanford MS&E211 Lecture Note#15...

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #15 1 Conic Linear Programming Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/˜yyye
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #15 2 Conic LP ( CLP ) minimize c x subject to a i x = b i , i = 1 , 2 , ..., m, x C, where C is a convex cone. Linear Programming (LP) : c , a i , x ∈ R n and C = R n + Second-Order Cone Programming (SOCP) : c , a i , x ∈ R n and C = SOC Semidefinite Programming (SDP) : c , a i , x ∈ M n and C = M n + Note that cone C can be a product of many (different) convex cones.
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #15 3 LP, SOCP and SDP Examples minimize 2 x 1 + x 2 + x 3 subject to x 1 + x 2 + x 3 = 1 , ( x 1 ; x 2 ; x 3 ) 0 . minimize 2 x 1 + x 2 + x 3 subject to x 1 + x 2 + x 3 = 1 , x 2 2 + x 2 3 x 1 . minimize 2 x 1 + x 2 + x 3 subject to x 1 + x 2 + x 3 = 1 , x 1 x 2 x 2 x 3 0 ,
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #15 4 where for SDP: c = 2 . 5 . 5 1 and a 1 = 1 . 5 . 5 1 . Example of Mixed Cones: minimize 2 x 1 + x 2 + x 3 + 2 x 4 + x 5 + x 6 subject to x 1 + x 2 + x 3 + x 4 + x 5 + x 6 = 1 , ( x 1 ; x 2 ; x 3 ) 0 , ( x 4 ; x 5 ; x 6 ) SOCP.
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #15 5 Convex Optimization or Convex Programming Convex Optimization : minimize a convex function over a convex constraint set/region. Note that the objective function need only to be convex within the feasible region. An important fact for CO: any local minimizer is a global minimizer. ( CO ) minimize c 0 ( x ) subject to c i ( x ) b i , i = 1 , 2 , ..., m, where c i ( x ) , i = 0 , 1 , ..., m , are convex functions of x . Every convex optimization problem can be equivalently formulated a CLP!
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #15 6 Dual of Conic LP The dual problem to ( CLP ) minimize c x subject to a i x = b i , i = 1 , 2 , ..., m, x C. is ( CLD ) maximize b T y subject to m i y i a i + s = c , s C , where y ∈ R m are the dual variables, s is called the dual slack vector/matrix, and C is the dual cone of C . Theorem 1 (Weak duality theorem) c x b T y = x s 0 for any feasible x of (CLP) and ( y , s ) of (CLD).
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #15 7 Self-Dual Cones Frequently, C = C , that is, they are self-dual .
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