Efficient lifting methods for unstructured mixed integer programs with multiple constraints

Bo Zeng, Purdue University

Abstract

In this thesis, we introduce efficient lifting methods to generate strong cutting planes for unstructured mixed integer programs (MIPs) with multiple constraints. Our results include improved sequential lifting methods and novel sequence independent lifting methods using multidimensional superadditive lifting functions. First we investigate sequential lifting for general 0–1 MIPs. We introduce a new scheme to obtain strong bounds on sequential lifting functions for general seed inequalities. This scheme yields a systematic procedure to derive the maximal set of a general seed inequality. This result can significantly reduce the computational burden associated with sequential lifting. Second, we perform a polyhedral study of the 0–1 knapsack problem with disjoint cardinality constraints (MCKP). This model generalizes many MIPs with multiple constraints and is a first step in the study of general unstructured 0–1 MIP sets. First, we derive a family of generalized cover inequalities (GCIs) that are strong for a restriction of the problem. We then derive a family of facet-defining inequalities for MCKP by polynomially lifting GCIs. We also characterize their maximal sets. Further, we develop a family of multidimensional superadditive lifting functions of provably high quality to perform sequence independent lifting for GCIs in MCKP. Third, we propose a general framework to construct new high-dimensional superadditive lifting functions using known lower-dimensional ones. This framework is practically significant since it yields a simple procedure to build multidimensional superadditive lifting functions. We illustrate its strength and versatility by building high-dimensional superadditive approximations for two generic MIP sets. The superadditive lifting functions developed in this thesis are the first multidimensional superadditive functions obtained in the literature. Finally, we perform a computational study of cutting planes generated from multidimensional superadditive lifting functions for 0–1 mixed integer programs with cardinality and precedence constraints. Our results show that even a small number of cutting planes generated from multiple problem rows often lead to a significant improvement in the solution of 0–1 integer programs as compared to state-of-the-art commercial solvers. This observation indicates that cutting planes generated from multiple rows can be effective in the solution of complex MIPs.

Degree

Ph.D.

Advisors

Richard, Purdue University.

Subject Area

Operations research

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