The presence of symmetry is common in certain types of scheduling problems. Symmetry can occur when one is scheduling a collection of jobs on multiple identical machines, or if one is determining production schedules for identical machines. General symmetry-breaking methods can be strengthened by taking advantage of the special structure of the symmetry group in … Read more
This paper investigates the role of dual information on the performances of heuristics designed for solving the set covering problem. After solving the linear programming relaxation of the problem, the dual information is used to obtain the two main approaches proposed here: (i) The size of the original problem is reduced and then the resulting … Read more
We show that half-integral polytopes obtained as the convex hull of a random set of half-integral points of the 0/1 cube have rank as high as Ω(logn/loglogn) with positive probability — even if the size of the set relative to the total number of half-integral points of the cube tends to 0. The high rank … Read more
We describe a computationally effective method for generating disjunctive inequalities for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs, and in the limit will generate an inequality as strong as can be produced by the cut-generating nonlinear program suggested by Stubbs and Mehrotra. Using this procedure, we … Read more
Proportional symbol maps are a cartographic tool to assist in the visualization and analysis of quantitative data associated with specific locations, such as earthquake magnitudes, oil well production, and temperature at weather stations. As the name suggests, symbol sizes are proportional to the magnitude of the physical quantities that they represent. We present two novel … Read more
The question as to whether the Gomory-Chvatal closure of a non-rational polytope is a polytope has been a longstanding open problem in integer programming. In this paper, we answer this question in the affirmative, by combining ideas from polyhedral theory and the geometry of numbers. ArticleDownload View PDF
We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints which arise naturally in various real-world applications such as portfolio selection and subset selection in regression. We propose a new semidefinite program (SDP) approach for computing the “best” diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation. We … Read more
In this paper, we show that the Chatal-Gomory closure of a compact convex set is a rational polytope. This resolves an open question discussed in Schrijver 1980 and generalizes the same result for the case of rational polytopes (Schrijver 1980), rational ellipsoids (Dey and Vielma 2010) and strictly convex sets (Dadush et. al 2010). In … Read more
The feasibility pump (FP) [5,7] has proved to be a successful heuristic for finding feasible solutions of mixed integer linear problems (MILPs). FP was improved in [1] for finding better quality solutions. Briefly, FP alternates between two sequences of points: one of feasible so- lutions for the relaxed problem (but not integer), and another of … Read more
The lift-and-project closure is the relaxation obtained by computing all lift-and-project cuts from the initial formulation of a mixed integer linear program or equivalently by computing all mixed integer Gomory cuts read from all tableau’s corresponding to feasible and infeasible bases. In this paper, we present an algorithm for approximating the value of the lift-and-project … Read more