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Handout14 - Recap Optimality conditions for LP Math 171A Linear Programming Lecture 14 Farkas Lemma and its Implications Philip E Gill c 2011 LP

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Math 171A: Linear Programming Lecture 14 Farkas’ Lemma and its Implications Philip E. Gill c ± 2011 http://ccom.ucsd.edu/~peg/math171a Monday, February 7th, 2011 Recap: Optimality conditions for LP LP minimize x c T x subject to Ax b with A an m × n matrix, b an m -vector and c an n -vector. The vector x * is a solution of LP if and only if : (a) Ax * b (b) c = A T a λ * a for some vector λ * a 0, where A a is the active constraint matrix at x * UCSD Center for Computational Mathematics Slide 2/41, Monday, February 7th, 2011 Example: minimize 2 x 1 + x 2 subject to the constraints: constraint #1: x 1 + x 2 1 constraint #2: x 2 0 constraint #3: x 1 0 Written in the form min c T x subject to Ax b , we get c = 2 1 ! , A = 1 1 0 1 1 0 , b = 1 0 0 UCSD Center for Computational Mathematics Slide 3/41, Monday, February 7th, 2011 c a 1 c a 1 a 3 a 2 x 1 x 2 x * ¯ x
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At the point x * = 0 1 ! , the active set is A = { 1 , 3 } with A a = 1 1 1 0 ! , b a = 1 0 ! Solving for the Lagrange multipliers gives A T a λ a = c 1 1 1 0 ! λ a = 2 1 ! λ a = 1 1 ! 0 x * is optimal. UCSD Center for Computational Mathematics Slide 5/41, Monday, February 7th, 2011 At the point ¯ x = 1 0 ! , the active set is A = { 1 , 2 } with A a = 1 1 0 1 ! , b a = 1 0 ! Solving for the Lagrange multipliers gives A T a λ a = c 1 0 1 1 ! λ a = 2 1 ! λ a = 2 - 1 ! 6≥ 0 ¯ x is not optimal. UCSD Center for Computational Mathematics Slide 6/41, Monday, February 7th, 2011 Proof of Farkas’ lemma Result (Farkas’ Lemma) (A) c T p 0 for all p such that A a p 0 if and only if (B) c = A T a λ * a for some λ * a 0 UCSD Center for Computational Mathematics Slide 7/41, Monday, February 7th, 2011 Proof: We have two statements that we must show are equivalent: (A) c T p 0 for all p such that A T a p 0 (B) c = A T a λ * a for some λ * a 0 We must show that (A) (B) and (B) (A) UCSD Center for Computational Mathematics Slide 8/41, Monday, February 7th, 2011
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It is “EASY” to show that (B) (A) If (B) holds, then c = A T a λ * a for some λ * a 0. Then, c T p = ( A T a λ * a ) T p = ( λ * a ) T ( A a p ) 0 for all p with A a p 0 c T p 0 for all p with A a p 0 (A) holds. UCSD Center for Computational Mathematics
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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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Handout14 - Recap Optimality conditions for LP Math 171A Linear Programming Lecture 14 Farkas Lemma and its Implications Philip E Gill c 2011 LP

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