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

Abstract

To cite this version:Florent Cadoux. Computing deep facet-defining disjunctive cuts for mixed-integer programming.[ Abstract: 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.frCoupes disjonctives profondes exposant des facettes pour la programmation entière générale Résumé : Le problème de séparation consisteà 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 peutêtre 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 pourraitêtre utilisé pour calculer des coupes profondes exposant une facette.

Two-Term Disjunctions on the Second-Order Cone

“…Since then, disjunctive cuts have been studied extensively in mixed integer linear and nonlinear optimization [6,30,7,19,29,17,24,16]. Chvátal-Gomory, lift-and-project, mixed-integer rounding (MIR), and split cuts are all special types of disjunctive cuts.…”

Local Cuts and Two-Period Convex Hull Closures for Big-Bucket Lot-Sizing Problems

“…As we illustrate in Section 5 in our computational results, using the Euclidean distance function often seems to result in better convergence than using the infinity norm. Moreover, we make a remark on the recent work of Cadoux (2010), who presented a proof that using 2-norm can provide "deepest disjunctive cuts" (rather than using standard linear norms). Although that work focuses on separating fractional points from a disjunctive polyhedron, this assertion seems in line with our computational experience, where the 2-norm often achieves the fastest converging cuts.…”

A computational study of the cutting plane tree algorithm for general mixed-integer linear programs

“…To truncate the feasible set and obtain a bounded solution, various normalization constraints have been studied in the literature (Balas, 1979, Balas and Bonami, 2009, Balas and Perregaard, 2002, Cadoux, 2010, Cornuéjols and Lemaréchal, 2006, Fischetti et al, 2009, Rey and Sagastizábal, 2002. The introduction of the normalization constraint may generate extreme points of CGLP that do not correspond to facets of the convex hull (closure) of the disjunctive set.…”

“…It turns out that the choice of the norm used in defining the projection is critical to the success of a computer implementation of the CGLP. Computational experiments reported in Cadoux (2010) suggest that cuts which minimize the 2 norm do not lead to an effective computational approach. We have confirmed this conclusion in our own experiments.…”

“…As a result, such cuts may be referred to as "deep" cuts (Cornuéjols and Lemaréchal, 2006, Cadoux, 2010, Rey and Sagastizábal, 2002. As one can recognize, this idea can be traced back to the very origins of optimization, i.e., separating hyperplane theorems.…”