f x m X i 1 p i G i x 0 n 1 5 G x p 6 Here indicates the complementarity of two

F x m x i 1 p i g i x 0 n 1 5 g x p 6 here indicates

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f ( x ) + m X i =1 p i G i ( x ) = 0 n × 1 (5) 0 G ( x ) p 0 . (6) Here ‘ ’ indicates the complementarity of two vectors. If vectors a R m and b R m have the same dimensions, then 0 a b 0 means that: (i) a 0 , (ii) b 0 , and (iii) a > b = 0 . Two vectors that satisfy condition (iii) are said to be complementary or perpendicular to each other. Lizhi Wang ([email protected]) IE 534 Linear Programming September 7, 2012 12 / 18
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KKT conditions for linear programming For linear programming, KKT conditions are both sufficient and necessary for optimality For any given x 0 , it is an optimal solution to the LP if and only if there exists p 0 that satisfies the KKT conditions Lizhi Wang ([email protected]) IE 534 Linear Programming September 7, 2012 13 / 18
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KKT conditions for nonlinear programming For general nonlinear programming, KKT conditions are neither sufficient nor necessary for optimality Solution ( x 1 = 1 , x 2 = 0) does not satisfy KKT conditions, but it is optimal to max x 1 s . t . x 2 + ( x 1 - 1) 3 0 x 2 0 . Solution ( x 1 = 0 . 5 , x 2 = 0 . 25) satisfies KKT conditions, but it is not optimal to min x 2 s . t . x 2 1 - x 1 + x 2 0 0 x 1 2 . Lizhi Wang ([email protected]) IE 534 Linear Programming September 7, 2012 14 / 18
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KKT conditions for the standard form LP max { c > x : Ax b, x 0 } max f ( x ) = c > x : G ( x ) = - A m × n I n × n x + b m × 1 0 n × 1 0 ( m + n ) × 1 f ( x ) + m X i =1 p i G i ( x ) = 0 n × 1 (7) 0 G ( x ) p 0 . (8) c + - A > I y μ = 0 (9) 0 b - Ax x y μ 0 (10) 0 b - Ax x y A > y - c 0 . (11) Lizhi Wang ([email protected]) IE 534 Linear Programming September 7, 2012 15 / 18
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LP optimality condition 0 b - Ax x y A > y - c 0 .
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