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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 8

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online