Convexifying Multilinear Sets with Cardinality Constraints: Structural Properties, Nested Case and Extensions

TL;DR

This paper explores the convexification of multilinear sets with cardinality constraints, providing structural insights, tractable convex hull descriptions under certain conditions, and extending these results to more general cases.

Contribution

It introduces properness conditions for multilinear terms, characterizes convex hulls with nested structures, and develops polynomial-time separation algorithms for the inequalities.

Findings

Convex hull description is tractable under properness conditions.

Explicit polyhedral description for nested multilinear terms.

Polynomial-time separation of inequalities for the convex hull.

Abstract

The problem of minimizing a multilinear function of binary variables is a well-studied NP-hard problem. The set of solutions of the standard linearization of this problem is called the multilinear set. We study a cardinality constrained version of it with upper and lower bounds on the number of nonzero variables. We call the set of solutions of the standard linearization of this problem a multilinear set with cardinality constraints. We characterize a set of conditions on these multilinear terms (called properness) and observe that under these conditions the convex hull description of the set is tractable via an extended formulation. We then give an explicit polyhedral description of the convex hull when the multilinear terms have a nested structure. Our description has an exponential number of inequalities which can be separated in polynomial time. Finally, we generalize these inequalities to obtain valid inequalities for the general case.