Maximal Lattice-free Sets Research Articles

The (not so) trivial lifting in two dimensions

When generating cutting-planes for mixed-integer programs from multiple rows of the simplex tableau, the usual approach has been to relax the integrality of the non-basic variables, compute an intersection cut, then strengthen the cut coefficients corresponding to integral non-basic variables using the so-called trivial lifting procedure. Although of polynomial-time complexity in theory, this lifting procedure can be computationally costly in practice. For the case of two-row relaxations, we present a practical algorithm that computes trivial lifting coefficients in constant time, for arbitrary maximal lattice-free sets. Computational experiments confirm that the algorithm works well in practice.

A proof of Lovász’s theorem on maximal lattice-free sets

Let K be a maximal lattice-free set in $${\mathbb{R}^d}$$ , that is, K is convex and closed subset of $${\mathbb{R}^d}$$ , the interior of K does not contain points of $${\mathbb{Z}^d}$$ and K is inclusion-maximal with respect to the above properties. A result of Lovasz asserts that if K is d-dimensional, then K is a polyhedron with at most 2 d facets, and the recession cone of K is a linear space spanned by vectors from $${\mathbb{Z}^d}$$ . A first complete proof of mentioned Lovasz’s result has been published in a paper of Basu, Conforti, Cornuejols and Zambelli (where the authors use Dirichlet’s approximation as a tool). The aim of this note is to give another proof of this result. Our proof relies on Minkowki’s first fundamental theorem from the geometry of numbers. We remark that the result of Lovasz is relevant in integer and mixed-integer optimization.

Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra

In this paper, we study the relationship between 2D lattice-free cuts, the family of cuts obtained by taking two-row relaxations of a mixed-integer program (MIP) and applying intersection cuts based on maximal lattice-free sets in R 2 , and various types of disjunctions. Recently, Li and Richard (2007) studied disjunctive cuts obtained from t-branch split disjunctions of mixed-integer sets (these cu ts generalize split cuts). Balas (2009) initiated the study of cuts for the two-row continuous group relaxation obtained from 2-branch split disjunctions. We study these cuts (and call them cross cuts) for the two-row continuous group relaxation, and for general MIPs. We also consider cuts obtained from asymmetric 2-branch disjunctions which we call crooked cross cuts. For the two-row continuous group relaxation, we show that unimodular cross cuts (the coefficients of the two split inequalities fo rm a unimodular matrix) are equivalent to the cuts obtained from maximal lattice-free sets other than type 3 triangles. We also prove that all 2D lattice-free cuts and their S-free extensions ar e crooked cross cuts. For general mixed integer sets, we show that crooked cross cuts can be generated from a structured three-row relaxation. Finally, we show that for the corner relaxation of an MIP, every crooked cross cut is a 2D lattice-free cut.