MPC 2011, ISSUE 2

Mathematical Programming Computation, Volume 3, Issue 2, June 2011

A relax-and-cut framework for Gomory mixed-integer cuts

Matteo Fischetti, Domenico Salvagnin

Gomory mixed-integer cuts (GMICs) are widely used in modern branchand-cut codes for the solution of mixed-integer programs. Typically, GMICs are iteratively generated from the optimal basis of the current linear programming (LP) relaxation, and immediately added to the LP before the next round of cuts is generated. Unfortunately, this approach is prone to instability. In this paper we analyze a different scheme for the generation of rank-1 GMICs read from a basis of the original LP—the one before the addition of any cut.We adopt a relax-and-cut approach where the generated GMICs are not added to the current LP, but immediately relaxed in a Lagrangian fashion. Various elaborations of the basic idea are presented, that lead to very fast—yet accurate—variants of the basic scheme. Very encouraging computational results are presented, with a comparison with alternative techniques from the literature also aimed at improving the GMIC quality. We also show how our method can be integrated with other cut generators, and successfully used in a cut-and-branch enumerative framework.

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Mathematical Programming Computation, Volume 3, Issue 2, June 2011

MIPLIB 2010 - Mixed Integer Programming Library version 5

Thorsten Koch, Tobias Achterberg, Erling Andersen, Oliver Bastert, Timo Berthold, Robert E. Bixby, Emilie Danna, Gerald Gamrath, Ambros M. Gleixner, Stefan Heinz, Andrea Lodi, Hans Mittelmann, Ted Ralphs, Domenico Salvagnin, Daniel E. Steffy, Kati Wolter

This paper reports on the fifth version of theMixed Integer Programming Library. The miplib 2010 is the first miplib release that has been assembled by a large group from academia and from industry, all of whom work in integer programming. There was mutual consent that the concept of the library had to be expanded in order to fulfill the needs of the community. The new version comprises 361 instances sorted into several groups. This includes the main benchmark test set of 87 instances, which are all solvable by today’s codes, and also the challenge test set with 164 instances, many of which are currently unsolved. For the first time, we include scripts to run automated tests in a predefined way. Further, there is a solution checker to test the accuracy of provided solutions using exact arithmetic.

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