New computational approaches to the solution of mixed integer programs

Amar K Narisetty, Purdue University

Abstract

This thesis focuses on the derivation of improved computational schemes for the optimization of mixed integer programs (MIPs). These schemes are based on new theoretical results and are evaluated numerically to determine their practical usefulness. In this document, we first present the motivation for our research through practical examples. We then perform a literature review before formulating statements for the problems we propose to solve. Finally, we review the results we have obtained with respect to each of these research questions. First, for improving the optimization of 0-1 MIPs we seek to derive new strong valid cutting planes for 0-1 MIPs through lifting techniques. In particular, we describe five families of strong inequalities for 0-1 MIP problems. We show that these inequalities can be applied to the simplex tableaux of the LP relaxations of 0-1 MIPs but also can be applied directly to their formulations. The separation of these cuts from simplex tableaux is immediate. The separation from the formulation is more difficult. We therefore propose several heuristics for it. We then present the results of a computational study comparing the performance of these cuts on a family of randomly generated problems and on instances from the MIPLIB 2003 library. Second, we propose a new lifting scheme that yields strong valid inequalities for 0-1 MIPs. This scheme gives rise to inequalities that can be described in closed-form but is different from traditional superadditive lifting. A new family of inequalities is derived for MIPs using this approach. Third, we propose a new paradigm for generating cuts for general MIPs that uses partial enumeration. A benefit of this approach is that new integer feasible solutions to the problem can be found during the cut generation process. We present computational results evaluating the effectiveness of this approach. Fourth, we describe computational work that was done for improving SAS mixed integer linear programming solver. These include improvements to presolve, cuts and branching. Finally, we conclude this thesis with a summary of research contributions and directions for future work.

Degree

Ph.D.

Advisors

Richard, Purdue University.

Subject Area

Operations research

COinS