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

Venue: | Operations Research |

Citations: | 8 - 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 ...jective; this model can easily be transformed into the one we use in the present study.) The raw data from which we have generated our instances has been taken from a public archive maintained at UCI =-=[33]-=-, and converted to the final MIP model with a slightly modified version of a program kindly provided us by P. A. Rubin. This resulted into 38 instances coming from different real situations (i.e., dis... |

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Citation Context ...thlike structure; other variations on this theme include dynamic backtracking, dependency-directed backtracking, partial-order dynamic backtracking, and generalized partial-order dynamic backtracking =-=[7, 14, 15, 16, 19]-=-. This nice feature is not preserved in our method, where the master problem is a (somehow general) NP-hard ILP. CB cuts can be generated within a standard Branch & Cut solution framework not based ex... |

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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 ... logic-based Benders’ cuts in [19] and proposed applying them to multiple machine scheduling, in which constraints hi(y) are scheduling constraints rather than linear inequalities. Jain and Grossmann =-=[26]-=- implemented this idea and obtained very good results—even without insisting on finding minimal conflict sets. Similar results were obtained for a broader class of problems in [20], but these results ... |

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Citation Context ...they are modelled. Actually, the approach suggests an extension of the MIP modelling language where logical implications of the type (2) can be stated explicitly in the model, as in Hooker and Osorio =-=[21]-=-. In this paper we aim at investigating whether the above method can be useful to approach certain types of MIP’s which are notoriously very hard to solve. As shown in the computational section, this ... |

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Citation Context ...he classical Benders’ method, our approach solves a single master problem and generate Benders’ cuts on the fly. This approach was proposed in [19] and later named “branch and check” by Thorsteinsson =-=[32]-=-, who successfully applied it to the Jain and Grossmann problems. Meanwhile, Hooker and Ottosson [22] used logic Benders’ cuts to solve SAT problems and 0-1 programming problems. Our solution method a... |

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Citation Context ... Indeed, in this case condition (27) can be rephrased as the CB cut � i∈C (1 − xj(i)) ≥ 1. Apparently, the first practical use of this kind of Benders’ cuts was to solve circuit verification problems =-=[23]-=-. In this application the subproblem constraints hi(y) are again linear inequalities (actually, systems of linear inequalities), but they have a special structure that permits the rapid identification... |

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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 ...thlike structure; other variations on this theme include dynamic backtracking, dependency-directed backtracking, partial-order dynamic backtracking, and generalized partial-order dynamic backtracking =-=[7, 14, 15, 16, 19]-=-. This nice feature is not preserved in our method, where the master problem is a (somehow general) NP-hard ILP. CB cuts can be generated within a standard Branch & Cut solution framework not based ex... |

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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... |

13 | A polyhedral study of the asymmetric traveling salesman problem with time windows
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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 ...ssification of all members, the problem amounts to deleting the minimum number of inequalities to make the system feasible, or equivalently to finding a maximum feasible subsystem (MaxFS); see, e.g., =-=[10, 1]-=-. 13sanimal behavior, etc.). 5.2 Results Our experiments have been performed on a PC AMD Athlon 2100+ with 1 GByte RAM, and GNU/Linux (kernel 2.4) Operating System. All the instances of our test-bed h... |

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Citation Context ...he form xij = 1 ⇒ yj ≥ yi + travel time(i, j) (4) whereas (3) bound the arrival time at each city i early arrival time(i) ≤ yi ≤ late arrival time(i). (5) Another example is the map labelling problem =-=[29]-=-, where the binary variables are associated to the relative position of two labels to be placed on a map, the continuous variables give their placement coordinates, and the conditional constraints imp... |

11 |
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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 ... 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 minimal infeasible co... |

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(Show Context)
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... |

9 |
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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 ... |

3 |
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Citation Context ... methods rather than using fast graph-search algorithms) is acceptable. Two-Group Statistical Classification (discriminant analysis). This problem can be described briefly as follow; see, e.g., Rubin =-=[31]-=- for more details. We have a population whose members can be divided into two distinct classes—for example, people affected or not by a certain disease. We can measure a number of characteristics that... |

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 ... |

1 |
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Citation Context ...he generated CB cuts, other classes of combinatorial cuts are likely to be violated, hence allowing other cut separators to obtain a further improvement. We found that the {0, 1 2 }–cuts addressed in =-=[9, 2]-=- fit particularly well in this framework, and contribute substantially to the overall efficacy of the approach. It is worth noting that, using the new technique, the role of the big-M terms in the MIP... |

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1 |
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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... |

1 |
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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... |