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mthsc810-lecture16-2x2

mthsc810-lecture16-2x2 - Sensitivity analysis MthSc 810...

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MthSc 810: Mathematical Programming Lecture 16 Pietro Belotti Dept. of Mathematical Sciences Clemson University November 3, 2011 Reading for today: Sections 5.1 Reading for Nov. 8: Sections 5.2, 5.3-5.5 Homework #9 due next Tuesday Homework #10 is out! Due next Thursday. Sensitivity analysis After solving P = min { c x : A x = b , x 0 } , we usually have an optimal basis B and optimal primal-dual solutions x , u . Is B still feasible/optimal if A , b , or c change? variables/constraints are added/eliminated? Depends on primal and dual optimality: does the new problem P = min { ˜c x : ˜ A x = ˜ b , x 0 } still have Primal feasibility: B 1 b 0 ? Dual feasibility: c N c B B 1 N 0 ? Variable added A = ( B | N ) ˜ A = ( ˜ B | ˜ N ) , where ... ˜ N = ( N | A n + 1 ) and ˜ B = B . ˜ c = ( c | c n + 1 ) . Unchanged b . For simplicity, let’s assume the new variable x n + 1 0. Primal feasibility: ˜ B 1 ˜ b = B 1 b 0 ? Of course. Dual feasibility: ˜ c N ˜ c B ˜ B 1 ˜ N 0 ? Maybe not. Dual feasibility requires ¯ c n + 1 = c n + 1 c B B 1 A n + 1 0.

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