Minimal Valid Inequalities for Integer Constraints

Abstract

In this note we consider an infinite relaxation of mixed integer linear programs. We show that any minimal valid inequality for this infinite relaxation arises from a nonnegative, piecewise linear, convex and homogeneous function.

Cut-Generating Functions and S-Free Sets

“…We have introduced for this the concept of cutgenerating functions, which allowed us to put in perspective an abundant literature devoted to S-free sets. We have revealed the discrepancy between minimality and maximal S-freeness; and we have recovered existing theorems [18,8,4,12,5], dealing with mere minimality, exhibiting the intrinsic arguments allowing their proofs. Our theory necessitated a generalization of the polarity correspondence to certain unbounded sets; we have conducted it via a systematic exploitation of the correspondence between sublinear functions and closed convex sets.…”

“…We are still in the context of (1) with q = m , R = −A , S = Z m − b ; this is the model considered in [1] for m = 2, and in [8] for general m. Other relevant references are [4,5,12,15,18].…”

“…(a) when S is a finite set of points in Z q − b; see [18] and more recently [12]; (b) when S is the intersection of Z n with an affine space; this was considered in [8] and [4]; (c) when S = P ∩ (Z q − b) for some rational polyhedron P ; this was considered in [12,5].…”

“…The aim of the paper is to present a formal theory of minimal cutgenerating functions and maximal S-free sets, valid independently of the particular S. Such a theory would gather and synthetize a number of papers dealing with the above problem for various special forms for S: [20,1,8,12,4,5] and references therein. For this, we use basic tools from convex analysis and geometry.…”

On the Practical Strength of Two-Row Tableau Cuts

“…Several variants of (2) have been studied; see for example [2,3,10,11,14,15,17,20,21,29] and [19] for a recent survey on the topic. This relaxation (2) has at least two appealing features : i) it is possible to obtain a complete characterization of all facet-defining inequalities using intersections cuts derived from the so-called maximal lattice-free convex sets and ii) the cutting planes derived from these facet-defining inequalities have the strongest possible coefficients for the continuous variables.…”