## Combinatorial Benders’ Cuts for Mixed-Integer Linear Programming

Venue: | Operations Research |

Citations: | 9 - 0 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}

}

### OpenURL

### 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.

### Citations

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Citation Context ...’ cut than (27). Cambazard et al. [8] returned to the idea of min-conflict Benders’ cuts to solve real-time scheduling problems on multiple machines. They relied on a min-conflict algorithm of Junker =-=[27]-=-, which improves on an earlier algorithm of De Siqueira and Puget [13]. Kim and Hooker [28] considered fixed-charge network flow problems, and proposed a scheme that is very similar to the approach pr... |

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Citation Context ...lict Benders’ cuts to solve real-time scheduling problems on multiple machines. They relied on a min-conflict algorithm of Junker [27], which improves on an earlier algorithm of De Siqueira and Puget =-=[13]-=-. Kim and Hooker [28] considered fixed-charge network flow problems, and proposed a scheme that is very similar to the approach proposed in this paper. The main difference is that Kim and Hooker find ... |

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Citation Context ...itional constraint aT i y ≤ bi in (2) by 1 − x∗ j(i) ; (ii) weigh each unconditional constraint in (3) by 0; and (iii) look for a minimum-weight MIS of the resulting weighted system—a NP-hard problem =-=[1, 17]-=-. A simple polynomial-time heuristic for CB-cut separation is as follows. Given the (fractional or integer) point x∗ to be separated, start with C := {i ∈ I : x∗ j(i) = 1}, verify the infeasibility of... |

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Citation Context ...), where CB cuts can only state the single infeasibility of certain paths and are dominated by much stronger classes of problem-specific inequalities, including the tournament inequalities studied in =-=[3, 4]-=-. However, our method proved to have some merits in handling difficult MIP problems where CB cuts play a role in describing in a strong polyhedral way the underlying combinatorial structure. Examples ... |

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Citation Context ...), where CB cuts can only state the single infeasibility of certain paths and are dominated by much stronger classes of problem-specific inequalities, including the tournament inequalities studied in =-=[3, 4]-=-. However, our method proved to have some merits in handling difficult MIP problems where CB cuts play a role in describing in a strong polyhedral way the underlying combinatorial structure. Examples ... |

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Citation Context ...e an efficient algorithm to find a MIS of an infeasible linear system, that fits particularly well within our solution approach. The method is in the spirit of the one discussed by Parker and Ryan in =-=[30]-=- MIS search can be formulated as follows: given an infeasible system of inequalities, say Ãy ≥ ˜ b, find an inclusion-minimal set of its rows yielding an infeasible system. We therefore construct the ... |

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Citation Context ... finding minimal conflict sets. Similar results were obtained for a broader class of problems in [20], but these results required a more sophisticated form of Benders’ cut than (27). Cambazard et al. =-=[8]-=- returned to the idea of min-conflict Benders’ cuts to solve real-time scheduling problems on multiple machines. They relied on a min-conflict algorithm of Junker [27], which improves on an earlier al... |

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2 |
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Citation Context ...n nodes, while they becomes relevant only when the corresponding x j(i) attains value 1 (typically, because of branching). Of course, we can get rid of the y variables by using Benders’ decomposition =-=[5]-=-, but the resulting cuts are weak and still depend on the big-M values. As a matter of fact, the classical Benders’ approach can be viewed as a tool to speed-up the solution of the LP relaxation, but ... |

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Citation Context ....1 (at least, on some problem classes which fit particularly well in our scheme). Finally, some conclusions are drawn in Section 6. The present paper is based on the master thesis of the first author =-=[12]-=-, which was awarded the 2003 Camerini-Carraresi prize by the Italian Operation Research association (AIRO). Moreover, the paper was presented at the IPCO X meeting held in New York, June 2004. 4s2 Com... |

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Citation Context ...rrent node NODEh.) Our approach can therefore be viewed as a simple (yet hopefully effective) method for “branching resequencing” in the spirit of the Dynamic Branch and Bound of Glover and Tangedhal =-=[18]-=-. 4 Fast MIS search In this section we describe an efficient algorithm to find a MIS of an infeasible linear system, that fits particularly well within our solution approach. The method is in the spir... |

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Citation Context ...itude with respect to ILOG-Cplex 8.1, one of the best MIP solvers on the market. Our technique is based on Hooker’s idea of deriving Benders’ cuts from minimal sets of inconsistencies, as proposed in =-=[19]-=-. In this respect, our main contributions have been (a) to present a separation heuristic that finds a minimal Benders’ cuts for the special case of conditional constraints with linear implications, a... |