Strong cutting planes for unstructured mixed integer programs using multiple constraints

Santanu S Dey, Purdue University

Abstract

In this thesis, we develop efficient methods to generate cutting planes for unstructured mixed integer programs using the information contained in several constraints simultaneously. To this end, we extend group and lifting approaches for cutting plane generation to mixed integer programs with multiple constraints. First we study multi-dimensional group cuts. We derive properties of extreme valid functions for infinite group problems. We show that pure integer infinite group problems can have discontinuous extreme inequalities while infinite group relaxations of mixed integer programs can only have continuous extreme inequalities. These results are used to develop general tools for the study of two-dimensional group problems. We then introduce the first two known families of facet-defining inequalities for the two-dimensional group problem. The first family provides a theoretical explanation for the empirical success of aggregation-based cut-generation schemes. The second family illustrates that stronger coefficient for continuous variables can be obtained when generating group cuts based on multiple constraints. Although using two rows of the simplex tableau, the two families of cutting planes we propose can be generated in linear time and are computationally attractive. Numerical experiments indicate that these new families of cuts can reduce the integrality gap of mixed integer programs more than when using Gomory mixed integer cuts alone. Moreover these experiments show that the new cuts can extract information from multiple rows that is not extracted using aggregation. Finally, we extend some of these results to general m-dimensional group problems. Second we study lifting techniques for sets with multiple constraints. We present a method that computes approximate lifting coefficients efficiently since it uses the solution of no more than two linear programs. This is a substantial improvement to the straightforward approach which requires the solution of a non-linear mixed integer problem. Our lifting method is applied to Glover's primal algorithm, and the computational improvement in the algorithm's performance is studied. To the best of our knowledge this lifting procedure yields the best known computational performance for cutting-plane based primal algorithm for unstructured mixed integer programs.

Degree

Ph.D.

Advisors

Richard, Purdue University.

Subject Area

Industrial engineering|Operations research

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