One form of characterizing the expressiveness of a piecewise linear neural network is by the number of linear regions, or pieces, of the function modeled. We have observed substantial progress in this topic through lower and upper bounds on the maximum number of linear regions and a counting procedure. However, these bounds only account for … Read more
A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative mixed integer linear programming formulations. Our first formulation is based on casting a standard quadratic program … Read more
In this paper, we present a new stochastic mixed-integer linear programming model for the Stochastic Outpatient Procedure Scheduling Problem (SOPSP). In this problem, we schedule a day’s worth of procedures for a single provider, where each procedure has a known type and associated probability distribution of random duration. Our objective is to minimize the expectation … Read more
Parallel Alternating Criteria Search (PACS) relies on the combination of computer parallelism and Large Neighborhood Searches to attempt to deliver high quality solutions to any generic Mixed-Integer Program (MIP) quickly. While general-purpose primal heuristics are widely used due to their universal application, they are usually outperformed by domain-specific heuristics when optimizing a particular problem class. … Read more
We establish a general computational framework for Chvátal’s conjecture based on exact rational integer programming. As a result we prove Chvátal’s conjecture holds for all downsets whose union of sets contains seven elements or less. The computational proof relies on an exact branch-and-bound certificate that allows for elementary verification and is independent of the integer … Read more
We study single- and multiple-ratio robust fractional 0-1 programming problems (RFPs). In particular, this work considers RFPs under a wide-range of disjoint and joint uncertainty sets, where the former implies separate uncertainty sets for each numerator and denominator, and the latter accounts for different forms of inter-relatedness between them. First, we demonstrate that, unlike the … Read more
We introduce a novel and powerful approach for solving certain classes of mixed integer programs (MIPs): decomposition branching. Two seminal and widely used techniques for solving MIPs, branch-and-bound and decomposition, form its foundation. Computational experiments with instances of a weighted set covering problem and a regionalized p-median facility location problem with assignment range constraints demonstrate … Read more
In this paper, we consider polytopes P that are contained in the unit hypercube. We provide conditions on the set of 0,1 vectors not contained in P that guarantee that P has a small Chvatal rank. Our conditions are in terms of the subgraph induced by these infeasible 0,1 vertices in the skeleton graph of … Read more
We study the following problem: given a rational polytope with Chvatal rank 1, does it contain an integer point? Boyd and Pulleyblank observed that this problem is in the complexity class NP ∩ co-NP, an indication that it is probably not NP-complete. It is open whether there is a polynomial time algorithm to solve the … Read more
Split cuts are prominent general-purpose cutting planes in integer programming. The split closure of a rational polyhedron is what is obtained after intersecting the half-spaces defined by all the split cuts for the polyhedron. In this paper, we prove that deciding whether the split closure of a rational polytope is empty is NP-hard, even when … Read more