Minimal Infeasible Subsystems and Benders cuts

There are many situations in mathematical programming where cutting planes can be generated by solving a certain “cut generation linear program” whose feasible solutions define a family of valid inequalities for the problem at hand. Disjunctive cuts and Benders cuts are two familiar examples. In this paper we concentrate on classical Benders cuts, as they belong to the basic toolbox for mixed-integer programming. It is a common experience however that the use of Benders cuts is not always as effective as hoped, especially if the impact of simple yet fundamental design issues are underestimated and the method is implemented “as in its textbook description”. In this paper we propose alternative selection criteria for Benders cuts, and analyze them computationally. Our approach is based on the correspondence between minimal infeasible subsystems of an infeasible LP, and the vertices of the so-called alternative polyhedron. The choice of the “most effective” violated Benders cut then correspond to the selection of a suitable vertex of the alternative polyhedron, hence a clever choice of the dual objective function is crucial—whereas the textbook Benders approach uses a completely random selection policy, at least when the so-called feasibility cuts are generated. Computational results on a testbed of MIPLIB instances are presented, where the quality of Benders cuts is measured in terms of “percentage of gap closed” at the root node, as customary in cutting plane methods. We show that the proposed methods allow for a speedup of 1 to 2 orders of magnitude with respect to the textbook one.