lscale - EE236A(Fall 2007-08 Lecture 16 Large-scale linear...

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 16 Large-scale linear programming • cutting-plane method • Benders decomposition • delayed column generation • Dantzig-Wolfe decomposition 16–1 Cutting-plane method minimize c T x subject to Ax ≤ b A ∈ R m × n , m ≫ n general idea : solve sequence of relaxations minimize c T x subject to a T i x ≤ b i , i ∈ I ⊆ { 1 , . . . , m } • gives a lower bound on the optimal value of the original problem • if x solves the relaxed LP and Ax ≤ b , then x solves the original LP • if x solves the relaxed LP and a T j x > b j for some j , then we add j to I and solve the new relaxed LP key to an efficient implementation: find a violated inequality without testing all inequalities Large-scale linear programming 16–2 Robust linear programming minimize c T x subject to ( a i + B i y ) T x ≤ b i for all y ∈ P and i = 1 , . . . , m where B i ∈ R n × p , P = { y | Cy ≤ d } assume P is bounded with extreme points y k , k = 1 , . . . , K ; problem is equivalent to minimize c T x subject to ( a i + B i y k ) T x ≤ b i for k = 1 , . . . , K and i = 1 , . . . , m • a linear program in x • a huge number of inequalities (except for very small p , or special choice of C , d ) Large-scale linear programming 16–3 key step in cutting-plane method: given x , find y k and i for which the inequality ( a i + B i y k ) T x ≤ b i is violated (and do this without enumerating all y k ) solution : solve the LPs (with variable y ) maximize x T B i y subject to Cy ≤ d let y ⋆ be an optimal extreme point • if x T B i y ⋆ > b i − a T i x , then y ⋆ defines a violated inequality • otherwise, ( a i + B i y ) T x ≤ b i for all y ∈ P very fast if dimension of C is small Large-scale linear programming 16–4 summary (for simplicity we add bounds l ≤ x ≤ u ) set N i = ∅ , i = 1 , . . . , m 1. let x...
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This note was uploaded on 09/26/2009 for the course CAAM 236 taught by Professor Dr.vandenber during the Spring '07 term at Monmouth IL.

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lscale - EE236A(Fall 2007-08 Lecture 16 Large-scale linear...

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