In optimization-based classification model selection, for example when using linear programming formulations, a standard approach is to penalize the L1 norm of some linear functional in order to select sparse models. Instead, we propose a novel integer linear program for sparse classifier selection, generalizing the minimum disagreement hyperplane problem whose complexity has been investigated in … Read more
We propose a unifying framework for polyhedral approximation in convex optimization. It subsumes classical methods, such as cutting plane and simplicial decomposition, but also includes new methods, and new versions/extensions of old methods, such as a simplicial decomposition method for nondifferentiable optimization, and a new piecewise linear approximation method for convex single commodity network flow … Read more
The Sherali-Adams lift-and-project hierarchy is a fundamental construct in integer programming, which provides successively tighter linear programming relaxations of the integer hull of a polytope. We initiate a new approach to understanding the Sherali-Adams procedure by relating it to methods from computational algebraic geometry. Our main result is a refinement of the Sherali-Adams procedure that … Read more
In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems has been receiving increasing attention in the literature as it allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with … Read more
A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. Recently Andersen et al. (2007) and Cornuejols and Margot (2007) showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from … Read more
Recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized. In this paper, we consider a stronger relaxation of general mixed integer sets by allowing constraints, such as bounds, on the free integer variables in the continuous group relaxation. We generalize a number of results for the continuous … Read more
Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposition methods for many classes of … Read more
We propose an approach to linear optimization with recourse that does not involve a probabilistic description of the uncertainty, and allows the decision-maker to adjust the degree of robustness of the model while preserving its linear properties. We model random variables as uncertain parameters belonging to a polyhedral uncertainty set and minimize the sum of … Read more
This paper addresses a constrained two-dimensional (2D), non-guillotine restricted, packing problem, where a fixed set of small rectangles has to be packed into a larger stock rectangle so as to maximize the value of the rectangles packed. The algorithm we propose hybridizes a novel placement procedure with a genetic algorithm based on random keys. We … Read more
Abstract: We present new valid inequalities for 0-1 programming problems that work in similar ways to well known cover inequalities. Discussion and analysis of these cuts is followed by their revision and use in integer programming as a new generation of cuts that excludes not only portions of polyhedra containing noninteger points, also parts with … Read more