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handout15 - Math 171A: Linear Programming Lecture 15...

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Unformatted text preview: Math 171A: Linear Programming Lecture 15 Testing for Optimality Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Wednesday, February 9th, 2011 Recap: Farkas’ lemma c T p ≥ 0 for all p such that A a p ≥ if and only if c = A T a λ * a for some λ * a ≥ UCSD Center for Computational Mathematics Slide 2/30, Wednesday, February 9th, 2011 Farkas’ lemma states that if a feasible x * is optimal, then there is at least one λ * a ≥ 0 such that A T a λ * a = c Unfortunately, if x * is optimal there may be infinitely many vectors λ a such that A T a λ a = c . Many of which will have [ λ a ] s < 0 for some index s ! UCSD Center for Computational Mathematics Slide 3/30, Wednesday, February 9th, 2011 Consider an LP with objective vector c = 2 2 . Suppose that ¯ x is a vertex with active-set matrix A a = 1 2 1 3 1 1- 1 1 Observe that ¯ x is a degenerate vertex. The equations A T a λ a = c have infinitely many solutions . UCSD Center for Computational Mathematics Slide 4/30, Wednesday, February 9th, 2011 The equations A T a λ a = c are 1 1 1- 1 2 3 1 1 λ 1 λ 2 λ 3 λ 4 λ 5 = 2 2 For example, ¯ λ = solve(Aa’,c) gives ¯ λ = 1- 1 , but another solution is λ * = 2 2 ⇒ ¯ x is optimal. UCSD Center for Computational Mathematics Slide 5/30, Wednesday, February 9th, 2011 A FUNDAMENTAL QUESTION! How do we find a nonnegative λ * a hidden among the infinitely many solutions of A T a λ a = c ? The answer to the fundamental question. . . A feasible x * with active-set matrix A a is a solution of an LP if and only if there is a nonnegative basic solution of A T a λ a = c . Proof: Later! UCSD Center for Computational Mathematics Slide 6/30, Wednesday, February 9th, 2011 There are a finite number of basic solutions of A T a λ a = c ....
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.

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handout15 - Math 171A: Linear Programming Lecture 15...

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