Local Cuts in Discrete Optimization and Mixed-Integer Programming

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Solution algorithms for difficult discrete optimization problems often rely on problem-specific cutting-planes to improve the associated linear-programming relaxations. It is standard practice to look for cutting planes that match prescribed templates, where the templates are drawn exclusively from the set of linear inequalities that induce facets of the convex hull of the solution set for the problem. In the traveling salesman problem (TSP) research of Applegate, Bixby, Chvatal, and Cook, an alternative to this template paradigm for cutting planes was proposed. The new procedure is called local cuts and it played a crucial role in the solution of a set of large-scale TSP instances, including a 13,509-city example and a 15,112-city example. The goal of this project is to extend the local-cut procedure to other classes of discrete optimization problems and to general mixed-integer programming models. The local-cut procedure consists of selecting a family of linear mappings taking the original solution space down to one of low dimension, and selecting a super-set of the image that is accessible, that is, it is easy to optimize linear functions over the super-set. Using delayed column-generation, cutting planes for the accessible set are determined; these cuts are then mapped back to cutting planes in the original set of variables. This project will continue the investigation of local cuts for the TSP, but a major part of the work will be to develop the methodology for other classes of problems, including mixed-integer programming models, vehicle routing, steiner trees, and stable sets.Broader Impact. Discrete optimization is used to solve practical problems that involve choosing the best alternative from a field of possibilities; it has broad applications in nearly every segment of the economy. The local-cut techniques developed in the proposed project will permit the solution of larger, more-complex problem instances, allowing users to be more aggressive in building models to represent the details of their applications.