Multilinear Sets with Cardinality Constraints

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, and many valid inequalities for its convex hull are available in the literature. Motivated by a machine learning application, we study a cardinality constrained version of this problem 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.