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 functions 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 weighted-sum … 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

Combination Chemotherapy Optimization

Chemotherapy is one of the primary modalities of cancer treatment. Chemotherapy drug administration is a complex problem that often requires expensive clinical trials to evaluate potential regimens. One way to alleviate this burden and better inform future trials is to build reliable models for drug administration. Previous chemotherapy optimization models have mainly relied on optimal … Read more

Objective Selection for Cancer Treatment: An Inverse Optimization Approach

In radiation therapy treatment-plan optimization, selecting a set of clinical objectives that are tractable and parsimonious yet effective is a challenging task. In clinical practice, this is typically done by trial and error based on the treatment planner’s subjective assessment, which often makes the planning process inefficient and inconsistent. We develop the objective selection problem … Read more

Relating Single-Scenario Facets to the Convex Hull of the Extensive Form of a Stochastic Single-Node Flow Polytope

Stochastic mixed-integer programs (SMIPs) are a widely-used modeling paradigm for sequential decision making under uncertainty. One popular solution approach to solving SMIPs is to solve the so-called “extensive form” directly as a large-scale (deterministic) mixed-integer program. In this work, we consider the question of when a facet-defining inequality for the convex hull of a deterministic, … Read more