Relaxations and Cutting Planes for Linear Programs with Complementarity Constraints

TL;DR

This paper develops advanced relaxation and cutting plane techniques for linear programs with complementarity constraints, improving solution bounds by leveraging conflict graph structures and polytope formulations.

Contribution

It generalizes existing reformulation techniques to handle complex complementarity conditions and introduces new cutting planes derived from stable set and boolean quadric polytopes.

Findings

Enhanced linear relaxations reduce optimality gaps in practical problems.

New cutting planes improve solution quality and computational efficiency.

Method demonstrates effectiveness across multiple problem types.

Abstract

We study relaxations for linear programs with complementarity constraints, especially instances whose complementary pairs of variables are not independent. Our formulation is based on identifying vertex covers of the conflict graph of the instance and generalizes the extended reformulation-linearization technique of Nguyen, Richard, and Tawarmalani to instances with general complementarity conditions between variables. We demonstrate how to obtain strong cutting planes for our formulation from both the stable set polytope and the boolean quadric polytope associated with a complete bipartite graph. Through an extensive computational study for three types of practical problems, we assess the performance of our proposed linear relaxation and new cutting-planes in terms of the optimality gap closed.