Computing deep facet-defining disjunctive cuts for mixed-integer programming

The problem of separation is to find an affine hyperplane, or “cut”, that lies between the origin O and a given closed convex set Q in a Euclidean space. We focus on cuts which are deep for the Euclidean distance, and facet-defining. The existence of a unique deepest cut is shown and cases when it is decomposable as a combination of facet-defining cuts are characterized using the reverse polar set. When Q is a split polyhedron, a new description of the reverse polar is given. A theoretical successive projections algorithm is proposed that could be used to compute deep facet-defining split cuts. Key-words: integer programming, separation, cut generation, disjunctive cut, convex analysis, reverse polar ∗ florent.cadoux@inria.fr in ria -0 01 44 28 9, v er si on 2 2 M ay 2 00 7 Coupes disjonctives profondes exposant des facettes pour la programmation entière générale Résumé : Le problème de séparation consiste à trouver un hyperplan affine, ou “coupe”, situé entre l’origine O et un ensemble convexe fermé Q dans un espace euclidien. On s’intéresse aux coupes profondes au sens de la distance euclidienne, et qui exposent une facette. L’existence d’une unique coupe de profondeur maximale est prouvée, et les cas où elle peut être décomposée en combinaison de coupes exposant une facette sont caractérisés grâce au polaire inverse de Q. Quand Q est un polyèdre disjonctif, une nouvelle description du polaire inverse est donnée. Un algorithme théorique de projections successives est proposé, qui pourrait être utilisé pour calculer des coupes profondes exposant une facette. Mots-clés : programmation entière, séparation, génération de coupes, coupes disjonctives, analyse convexe, polaire inverse in ria -0 01 44 28 9, v er si on 2 2 M ay 2 00 7 Computing deep facet-defining cuts 3 Introduction The problem of separation is essential in combinatorial optimization, occuring for instance in the following context: a function must be optimized on a “complicated” set in R, typically a set of integer points; this problem being too hard, an easier relaxation is solved; the problem is then to separate the solution of the relaxation from the complicated set, so as to tighten the relaxation. Separating hyperplanes, called cuts in the community of combinatorial optimization, are used in many practical methods aimed at solving mixed integer programs, often embedded in a branch-and-bound framework like the branch-and-cut algorithm. An overview of the existing general techniques to compute such cuts can be found in [3]. We will focus on particular cuts, called split cuts which belong to the family of disjunctive cuts [1]. In [2], lift-and-project cuts are defined and shown to be a particular case of split cuts for the mixed 0-1 case. In [7], a procedure is devised to generate one facet-defining lift-and-project cut. In this paper, we address the problem of generating facet-defining cuts which are deep; besides, we consider the general case of split cuts for the mixed integer case. The key idea to address the general problem of separation between O and some convex set Q is to use a precise correspondence between facets of Q and extreme points of another convex set in the dual space: the reverse polar of Q. Moreover, a facet is deep in some sense when the corresponding extreme point of the reverse polar is close to the origin in the dual space. These two facts lead us to a theoretical algorithm aimed at computing deep facetdefining cuts by solving an optimization problem, more precisely a quadratic programming problem, on the reverse polar set. We then particularize the theory to the special case, of great practical importance, where the point to be separated is the solution of the linear relaxation of a mixed integer program, and the convex set is a disjunctive (split) polyhedron. The article is organized as follows: section 1 introduces our main object, the reverse polar of a convex set. Section 2 advocates deep, facet-defining cuts as a good choice for the applications. A characterization of such cuts is given which uses the projection of O onto the reverse polar Q and its decomposition as a convex combination of extreme points of Q. Section 3 is devoted to the problem of computing the projection of a point onto a closed convex set, and the above-mentionned decomposition. Section 4 gives a characterization of Q when Q is an explicitly described polyhedron, which shows that computing the projection of O onto Q is tractable in this case. Section 5 introduces disjunctive programming and section 6 generalizes the method of section 4 to the case where Q is a split polyhedron. We finally propose a theoretical algorithm to compute deep, facet-defining cuts split cuts for general mixed integer programs. RR n 6177 in ria -0 01 44 28 9, v er si on 2 2 M ay 2 00 7