compslack - OR 3300/5300 Optimization I Prof Bland Fall...

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OR 3300/5300 Optimization I Fall 2011 Prof. Bland Complementary Slackness Part I: Motivating Complementary Slackness Consider the linear programming problem (P): maximize 6 x 1 + 14 x 2 + 13 x 3 + 30 x 4 subject to 1 2 x 1 + 2 x 2 + x 3 + 3 x 4 + 2 x 5 24 - x 1 + 4 x 2 + 4 x 3 - x 4 10 x 1 + 6 x 2 - 3 x 3 + x 5 20 ( P ) x 1 + 2 x 2 + 4 x 3 - 2 x 4 + 3 x 5 60 x 1 + x 2 + x 3 + x 4 + x 5 50 2 x 1 + 3 x 2 - x 3 + x 4 + 2 x 5 70 x 1 0 , x 2 0 , x 3 0 , x 4 0 , x 5 0 Proposed solution: b x 1 = 36 , b x 2 = 0 , b x 3 = 6 , b x 4 = b x 5 = 0 We want to check whether this solution is optimal. First check that it is feasible (it is). Duality tells us that if b x is optimal, there must exist a dual feasible b y with b yb = c b x . Let’s examine the implications of that. A b x = 24 24 - 12 10 18 20 60 60 42 50 66 70 = b
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b x T = 36 0 6 0 0 c = 6 14 13 30 0 b yA = Part II: Complementary Slackness (Read 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
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