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

Theorems of the Alternative for Conic Integer Programming

Farkas’ Lemma is a foundational result in linear programming, with implications in duality, optimality conditions, and stochastic and bilevel programming. Its generalizations are known as theorems of the alternative. There exist theorems of the alternative for integer programming and conic programming. We present theorems of the alternative for conic integer programming. We provide a nested … Read more

The Gap Function: Evaluating Integer Programming Models over Multiple Right-hand Sides

For an integer programming model with fixed data, the linear programming relaxation gap is considered one of the most important measures of model quality. There is no consensus, however, on appropriate measures of model quality that account for data variation. In particular, when the right-hand side is not known exactly, one must assess a model … Read more