Lagrangian relaxation is commonly used to generate bounds for mixed-integer linear programming problems. However, when the number of dualized constraints is very large (exponential in the dimension of the primal problem), explicit dualization is no longer possible. In order to reduce the dual dimension, different heuristics were proposed. They involve a separation procedure to dynamically … Read more
Within the context of solving Mixed-Integer Linear Programs by a Branch-and-Cut algorithm, we propose a new strategy for branching. Computational experiments show that, on the majority of our test instances, this approach enumerates fewer nodes than traditional branching. On average, on instances that contain both integer and continuous variables the number of nodes in the … Read more
This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non-convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, … Read more
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 … Read more
Many industrial problems can be naturally formulated using Mixed Integer Nonlinear Programming (MINLP). Motivated by the demand for Open-Source solvers for real-world MINLP problems, we have developed a spatial Branch-and-Bound software package named COUENNE (Convex Over- and Under-ENvelopes for Nonlinear Estimation). In this paper, we present the structure of couenne and discuss in detail our … Read more
Given a power grid modeled by a network together with equations describing the power flows, power generation and consumption, and the laws of physics, the so-called N – k problem asks whether there exists a set of k or fewer arcs whose removal will cause the system to fail. We present theoretical results and computation … Read more
We prove that the subset sum problem has a polynomial time computable certificate of infeasibility for all $a$ weight vectors with density at most $1/(2n)$ and for almost all integer right hand sides. The certificate is branching on a hyperplane, i.e. by a methodology dual to the one explored by Lagarias and Odlyzko; Frieze; Furst … Read more
The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0-1 knapsack set. One motivation for studying the submodular knapsack polytope is to address 0-1 programming problems with uncertain coefficients. Under various assumptions, a probabilistic constraint on 0-1 variables can be modeled as … Read more
We extend work on generating relaxations of simple sets with guaranteed bounds. Citation Unpublished. Columbia University, July 2008. Article Download View Tightening simple mixed-integer sets with guaranteed bounds
We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in Kaibel and Pfetsch (Math. Program. 114 (1), 2008, 1-36). These polytopes are the convex hulls of all 0/1-matrices with lexicographically sorted columns and at most, resp. exactly, one 1-entry per row. They are … Read more