Multilinear Sets with Two Monomials and Cardinality Constraints

TL;DR

This paper studies the convex hull of solutions for a specific binary polynomial optimization problem involving two monomials and cardinality constraints, providing an extended formulation and an efficient separation algorithm.

Contribution

It introduces an extended formulation with auxiliary variables and inequalities for the convex hull in the case of two monomials, and demonstrates efficient separation.

Findings

Extended formulation with exponential inequalities

Efficient separation algorithm developed

Convex hull characterization for the two-monomial case

Abstract

Binary polynomial optimization is equivalent to the problem of minimizing a linear function over the intersection of the multilinear set with a polyhedron. Many families of valid inequalities for the multilinear set are available in the literature, though giving a polyhedral characterization of the convex hull is not tractable in general as binary polynomial optimization is NP-hard. In this paper, we study the cardinality constrained multilinear set in the special case when the number of monomials is exactly two. We give an extended formulation, with two more auxiliary variables and exponentially many inequalities, of the convex hull of solutions of the standard linearization of this problem. We also show that the separation problem can be solved efficiently.