New Classes of Facets for Complementarity Knapsack Problems

TL;DR

This paper introduces three new classes of cutting-planes for the complementarity knapsack problem, enhancing polyhedral understanding and providing efficient algorithms for their separation.

Contribution

It extends polyhedral studies of CKP by proposing new cutting-planes derived from pack concepts and establishes conditions for their facet-defining properties.

Findings

New families of cutting-planes based on pack concepts.

Conditions for inequalities to be facet-defining.

Efficient separation algorithms for the new inequalities.

Abstract

The complementarity knapsack problem (CKP) is a knapsack problem with real-valued variables and complementarity conditions between pairs of its variables. We extend the polyhedral studies of De Farias et al. for CKP, by proposing three new families of cutting-planes that are all obtained from a combinatorial concept known as a pack. Sufficient conditions for these inequalities to be facet-defining, based on the concept of a maximal switching pack, are also provided. Moreover, we answer positively a conjecture by de Farias et~al.~about the separation complexity of the inequalities introduced in their work, and propose efficient separation algorithms for our newly defined cutting-planes.