arXiv subjects
Publications and source records attributed to Gerard Cornuejols.
We show that, given a closed convex set $K$ containing the origin in its interior, the support function of the set $\{y\in K^*: \exists x\in K\mbox{ such that } \langle x,y \rangle =1\}$ is the pointwise smallest among all sublinear functions $σ$ such that $K=\{x: σ(x)\leq 1\}$.
Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of these three families of inequalities. In particular we study how well each family approximates the integer hull. We show that, in a well defined sense, triangle inequalities provide a good approximation of the integer hull. The same statement holds for quadrilateral inequalities. On the other hand, the approximation produced by split inequalities may be arbitrarily bad.
In Mathematical Programming 2003, Gomory and Johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions. In this paper we give an example showing that the Gomory-Johnson conjecture is false.
This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting. The answer is a nonconvex region that can be obtained as a finite union of convex polyhedra. We then establish a necessary and sufficient condition for the uniqueness of the lifting function.
We show that maximal $S$-free convex sets are polyhedra when $S$ is the set of integral points in some rational polyhedron of $\mathbb{R}^n$. This result extends a theorem of Lovász characterizing maximal lattice-free convex sets. Our theorem has implications in integer programming. In particular, we show that maximal $S$-free convex sets are in one-to-one correspondence with minimal inequalities.
There has been a recent interest in cutting planes generated from two or more rows of the optimal simplex tableau. One can construct examples of integer programs for which a single cutting plane generated from two rows dominates the entire split closure. Motivated by these theoretical results, we study the effect of adding a family of cutting planes generated from two rows on a set of instances from the MIPLIB library. The conclusion of whether these cuts are competitive with GMI cuts is very sensitive to the experimental setup. In particular, we consider the issue of reliability versus aggressiveness of the cut generators, an issue that is usually not addressed in the literature.
We consider a model that arises in integer programming, and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lovász characterizing maximal lattice-free convex sets in $\mathbb{R}^n$.
We consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cut generated from a bounded convex set $P$ is finite if and only if the integer points on the boundary of $P$ satisfy a certain "2-hyperplane property". The Dey-Louveaux characterization is a consequence of this more general result.