1 1 1 Lizhi Wang lzwangiastateedu IE 534 Linear Programming 11

# 1 1 1 lizhi wang lzwangiastateedu ie 534 linear

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1 1 0 0 0 0 0 - 1 0 0 0 Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 11 / 21
Outline 1 Brief introduction 2 Gomory cut Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 12 / 21
Cutting plane algorithm Step 1: Solve LP relaxation and go to Step 2. Step 2: if LP relaxation infeasible then 2(a) return ILP is infeasible. else if x L is integral then 2(b) return x L is optimal to ILP. else 2(c) add a cut and go to Step 1. end Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 13 / 21
Ralph Edward Gomory Ralph Edward Gomory (1923 – ). American applied mathematician and executive. President of the Sloan Foundation, 1989 – 2007. He proposed the Gomory cut method for ILP in the 1950s. I Most experts, including Gomory himself, considered them to be impractical. I In the mid-1990s, Gomory cuts were shown to be very effective in combination with branch-and-bound. I Nowadays, all commercial MILP solvers use Gomory cuts in one way or another. Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 14 / 21
Gomory cutSuppose an LP relaxation is solved, the optimal solutionx*doesnot satisfy the integrality constraint, and the optimal simplexdictionary is the following:ζ=c>BA-1Bb+ (c>N-c>BA-1BAN)xNxB=A-1Bb-A-1BANxN.Letx*ibe a variable that violates the integrality constraint, and letthe following equation denote the corresponding row in thedictionary:xi=ai,0-Xj∈Nai,jxj.Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 15 / 21
Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 16 / 21
Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 17 / 21

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