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handout10 - Recap: Optimality conditions for ELP Math 171A:...

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Math 171A: Linear Programming Lecture 10 Feasible Directions and Vertices Philip E. Gill c ± 2011 http://ccom.ucsd.edu/~peg/math171a Friday, January 28, 2011 Recap: Optimality conditions for ELP Linear programming with equality constraints: ELP minimize x R n c T x subject to Ax = b A point ¯ x is optimal if and only if A ¯ x = b and there is a vector λ satisfying A T λ = c The components of λ are called the Lagrange multipliers . UCSD Center for Computational Mathematics Slide 2/37, Friday, January 28, 2011 Recap: Checking optimality with Matlab x = solve(A,b) If incompatible, no feasible point exists If compatible, x is feasible lambda = solve(A’,c) If incompatible, no bounded solution If compatible, x is a minimizer Example 1: A = ± - 1 5 0 1 1 3 - 1 1 4 2 ² , b = ± 4 - 5 ² , c = - 4 6 - 1 - 3 - 1 rank( A ) = 2 A has independent rows . Feasibility? x * = solve(A,b) x * = 0 1 0 - 1 0 , which is a basic solution of Ax = b UCSD Center for Computational Mathematics Slide 4/37, Friday, January 28, 2011
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Example 1: (continued) A = ± - 1 5 0 1 1 3 - 1 1 4 2 ² , b = ± 4 - 5 ² , c = - 4 6 - 1 - 3 - 1 Optimality? λ * = solve(A’,c) λ * = ± 1 - 1 ² , which are unique Lagrange multipliers x * is optimal, with ( x * ) = c T x * = 9. UCSD Center for Computational Mathematics Slide 5/37, Friday, January 28, 2011 Example 1: (continued) A = ± - 1 5 0 1 1 3 - 1 1 4 2 ² , b = ± 4 - 5 ² , c = - 4 6 - 1 - 3 - 1 Note that b x = - 3 2 1 2 0 0 0 is another basic solution of Ax = b , with ( b x ) = c T b x = 9. UCSD Center for Computational Mathematics Slide 6/37, Friday, January 28, 2011 Example 2: A = ± - 1 5 0 1 1 3 - 1 1 4 2 ² , b = ± 4 - 5 ² , c = - 8 - 2 6 3 3 rank( A ) = 2 A has independent rows . Feasibility? x * = solve(A,b) x * = 0 1 0 - 1 0 , which is a basic solution of Ax = b UCSD Center for Computational Mathematics Slide 7/37, Friday, January 28, 2011 Example 2: (continued) A = ± - 1 5 0 1 1 3 - 1 1 4 2 ² , b = ± 4 - 5 ² , c = - 8 - 2 6 3 3 Optimality? λ * = solve(A’,c) gives an incompatibility message. c 6∈ range( A T ) and the solution is unbounded. UCSD Center for Computational Mathematics Slide 8/37, Friday, January 28, 2011
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Example 3: A = 0 0 5 1 5 2 4 - 1 6 0 4 - 1 1 3 0 4 2 2 , b = 5 1 2 - 1 - 1 - 2 , c = 1 5 7 rank( A ) = 3 A has independent columns .
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handout10 - Recap: Optimality conditions for ELP Math 171A:...

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