Advanced mixed-integer programming formulations : methodology, computation, and application

Abstract

This thesis introduces systematic ways to use mixed-integer programming (MIP) to solve difficult nonconvex optimization problems arising in application areas as varied as operations, robotics, power systems, and machine learning. Our goal is to produce MIP formulations that perform extremely well in practice, requiring us to balance qualities often in opposition: formulation size, strength, and branching behavior. We start by studying a combinatorial framework for building MIP formulations, and present a complete graphical characterization of its expressive power. Our approach allows us to produce strong and small formulations for a variety of structures, including piecewise linear functions, relaxations for multilinear functions, and obstacle avoidance constraints. Second, we present a geometric way to construct MIP formulations, and use it to investigate the potential advantages of general integer (as opposed to binary) MIP formulations. We are able to apply our geometric construction method to piecewise linear functions and annulus constraints, producing small, strong general integer MIP formulations that induce favorable behavior in a branch-and-bound algorithm. Third, we perform an in-depth computational study of MIP formulations for nonconvex piecewise linear functions, showing that the new formulations devised in this thesis outperform existing approaches, often substantially (e.g. solving to optimality in orders of magnitude less time). We also highlight how high-level, easy-to-use computational tools, built on top of the JuMP modeling language, can help make these advanced formulations accessible to practitioners and researchers. Furthermore, we study high-dimensional piecewise linear functions arising in the context of deep learning, and develop a new strong formulation and valid inequalities for this structure. We close the thesis by answering a speculative question: Given a disjunctive constraint, what can we reasonably sacrifice in order to construct MIP formulations with very few integer variables? We show that, if we allow our formulations to introduce spurious "integer holes" in their interior, we can produce strong formulations for any disjunctive constraint with only two integer variables and a linear number of inequalities (and reduce this further to a constant number for specific structures). We provide a framework to encompass these MIP-with-holes formulations, and show how to modify standard MIP algorithmic tools such as branch-and-bound and cutting planes to handle the holes.