Representing Integer Program Value Function with Neural Networks

We study the value function of an integer program (IP) by characterizing how its optimal value changes as the right-hand side varies. We show that the IP value function can be approximated to any desired degree of accuracy using machine learning (ML) techniques. Since an IP value function is a Chv├ítal-Gomory (CG) function, we first … Read more

Strategy Investments in Matrix Games

We propose an extension of matrix games where the row player may select rows and remove columns, subject to a budget constraint. We present an exact mixed-integer linear programming (MILP) formulation for the problem, provide analytical results concerning its solution, and discuss applications in the security domain. Our computational experiments show heuristic approaches on average … Read more

Strengthened MIP Formulations for the Liver Region Redesign Models of Akshat et al.

Liver transplantation has been a critical issue in the U.S. healthcare system for decades, and the region redesign aims to ameliorate this issue. This paper revisits two mixed integer programming (MIP) formulations of the liver region redesign problem proposed by Akshat et al. We study their first formulation considering two different modeling approaches: one compact … Read more

Markov Decision Process Design: A Framework for Integrating Strategic and Operational Decisions

We consider the problem of optimally designing a system for repeated use under uncertainty. We develop a modeling framework that integrates design and operational phases, which are represented by a mixed-integer program and discounted-cost infinite-horizon Markov decision processes, respectively. We seek to simultaneously minimize the design costs and the subsequent expected operational costs. This problem … Read more

Characterizing Rational Transplant Program Response to Outcome-Based Regulation

Organ transplantation is an increasingly common therapy for many types of end-stage organ failure, including lungs, hearts, kidneys and livers. The past twenty years have seen increased scrutiny of post-transplant outcomes in the United States, in order to ensure the efficient utilization of the scarce organ supply. Under regulations by the Organ Procurement Transplantation Network … Read more

Evaluating Mixed-Integer Programming Models over Multiple Right-hand Sides

A critical measure of model quality for a mixed-integer program (MIP) is the difference, or gap, between its optimal objective value and that of its linear programming relaxation. In some cases, the right-hand side is not known exactly; however, there is no consensus metric for evaluating a MIP model when considering multiple right-hand sides. In … Read more

Relaxations and Duality for Multiobjective Integer Programming

Multiobjective integer programs (MOIPs) simultaneously optimize multiple objective func- tions over a set of linear constraints and integer variables. In this paper, we present continuous, convex hull and Lagrangian relaxations for MOIPs and examine the relationship among them. The convex hull relaxation is tight at supported solutions, i.e., those that can be derived via a … Read more

Superadditive duality and convex hulls for mixed-integer conic optimization

We present an infinite family of linear valid inequalities for a mixed-integer conic program, and prove that these inequalities describe the convex hull of the feasible set when this set is bounded and described by integral data. The main element of our proof is to establish a new strong superadditive dual for mixed-integer conic programming … Read more

On strong duality, theorems of the alternative, and projections in conic optimization

A conic program is the problem of optimizing a linear function over a closed convex cone intersected with an affine preimage of another cone. We analyse three constraint qualifications, namely a Closedness CQ, Slater CQ, and Boundedness CQ (also called Clark- Duffin theorem), that are sufficient for achieving strong duality and show that the first … Read more

Approximate Submodularity and Its Implications in Discrete Optimization

Submodularity, a discrete analog of convexity, is a key property in discrete optimization that features in the construction of valid inequalities and analysis of the greedy algorithm. In this paper, we broaden the approximate submodularity literature, which so far has largely focused on variants of greedy algorithms and iterative approaches. We define metrics that quantify … Read more