Progress in computational mixed integer programming—A look back from the other side of the tipping point

Abstract

The last few years have been a thrilling time for the commercial application of mixed integer programming. The technology has gone through an inflection point. Just a few years ago, MIP was viewed as a temptingly powerful modeling paradigm that would consistently disappoint in practice. In constrast, in the last few years MIP has become a vital capability that powers a wide range of applications in a variety of application domains. The shift can clearly be seen in the views of its practitioners, past and present. At one extreme, in the past year we have personally interacted with several OR practitioners who have not had direct exposure to modern MIP technology in several years. Their near-universal reaction to the claim that MIP models of useful size and complexity can now be solved is disbelief. Convincing them of the practicality of MIP modeling can require multiple, corroborating witnesses. At the other extreme, the expectations of users with recent experience can be quite surprising for those who have worked in the field for a while. A sophisticated MIP modeler recently expressed disappointment when models with millions of constraints and hundreds of thousands of binary variables could not be solved to optimality overnight. He had to console himself with solutions with 5% optimality gaps. The technology has clearly come a long way.One of the most remarkable observations about the transformation that has occurred in the last few years is that, for the most part, it was enabled by methods and theory that have been available in the academic literature for decades. We find this fact remarkable for two reasons. The first is that these insights and techniques, which were developed in the context of the then available MIP models, have proven so effective on the wide range of practical models solved today. Despite the generality and flexibility of the MIP modeling paradigm, the fundamental character of the models and the sources of inherent difficulty have not changed significantly with time. The second reason is simply that these techniques had not made it into commercial MIP software, despite their effectiveness.

Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex Generalized Disjunctive Programs

“…The (BM) and (HR) reformulations of this example are shown in (18) and (19) respectively. Note that in (18) we are additionally relaxing some of the equality constraints as inequalities, in order to avoid additional constraints.…”

“…Note that in (18) we are additionally relaxing some of the equality constraints as inequalities, in order to avoid additional constraints. It is easy to show that the problem with inequality relaxations is equivalent to the original problem.…”

Cut-Generating Functions and S-Free Sets

“…One of the most well-known cut-generating functions in integer programming is the so-called Gomory function [13], which we presented in Examples 1.1 and 1.2. The corresponding cuts can be generated quickly, so they are a powerful tool in computations; indeed, they drastically speed up integer-programming solvers [7].…”

Computational Study of Cutting Planes for a Lot-Sizing Problem in Branch-and-Cut Algorithm

“…According to the study of Balas et al [3] and Bixby and Rothberg [5], GMIC turns out to be the most effective cutting plane in practice. In order to generate a GMIC, we first solve LP relaxation and find a basic variable with a fractional value from the simplex tableau.…”