Combinatorial Benders’ Cuts for Mixed-Integer Linear Programming
| Venue: | Operations Research |
| Citations: | 21 - 1 self |
BibTeX
@ARTICLE{Codato_combinatorialbenders’,
author = {Gianni Codato and Matteo Fischetti},
title = {Combinatorial Benders’ Cuts for Mixed-Integer Linear Programming},
journal = {Operations Research},
year = {},
volume = {54},
pages = {2006}
}
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Abstract
Mixed-Integer Programs (MIP’s) involving logical implications modelled through big-M coefficients, are notoriously among the hardest to solve. In this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the big-M coefficients. Our solution scheme defines a master Integer Linear Problem (ILP) with no continuous variables, which contains combinatorial information on the feasible integer variable combinations that can be “distilled ” from the original MIP model. The master solutions are sent to a slave Linear Program (LP), which validates them and possibly returns combinatorial inequalities to be added to the current master ILP. The inequalities are associated to minimal (or irreducible) infeasible subsystems of a certain linear system, and can be separated efficiently in case the master solution is integer. The overall solution mechanism resembles closely the Benders ’ one, but the cuts we produce are purely combinatorial and do not depend on the big-M values used in the MIP formulation. This produces an LP relaxation of the master problem which can be considerably tighter than the one associated with original MIP formulation. Computational results on two specific classes of hard-to-solve MIP’s indicate the new method produces a reformulation which can be solved some orders of magnitude faster than the original MIP model.
Keyphrases
mixed-integer linear programming combinatorial bender cut original mip model big-m coefficient master solution original mip formulation specific class big-m value general applicability lp relaxation mip formulation certain linear system new method master integer linear problem overall solution mechanism automatic problem reformulation continuous variable feasible integer variable combination slave linear program combinatorial information return combinatorial inequality computational result solution scheme current master ilp mixed-integer program infeasible subsystem master problem hard-to-solve mip model dependency logical implication
