compslack - OR 320/520 Optimization I Prof. Bland 10/30/07...

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OR 320/520 Optimization I 10/30/07 Prof. Bland Complementary Slackness see BHM section 4.5 Consider a standard form linear programming problem (P) and its dual (D): max cx min yb ( P ) s.t. Ax b ( D ) s.t. yA c x 0 y 0 Assume that A is m × n A pair of vectors x = ( x 1 , ..., x n ) T , y = ( y 1 , ..., y m ) is complementary if (a) y i = 0 or n j =1 a ij x j = b i for i = 1 , ..., m and (b) x j = 0 or m i =1 a ij y i = c j for j = 1 , ..., n. The m + n conditions (a) and (b) are called complementarity conditions, or com- plementary slackness conditions. Note that (a) = yAx = yb (b) = yAx = cx . Hence, if the pair x, y is complementary, then cx = yb . Note if we introduce slack variables x n +1 ...x n + m in (P) and introduce surplus variables y m +1 , ..., y m + n in (D), the complementarity conditions (a) and (b) become: (a) y i = 0 or x n + i = 0 i = 1 , ..., m (b) x j = 0 or y m + j = 0 j = 1 , ..., n Complementary Slackness Theorem 1. If ( b x, b y ) is complementary, then
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compslack - OR 320/520 Optimization I Prof. Bland 10/30/07...

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