Cardinality Constrained Multilinear Sets.

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 cardinality constrained multilinear set, and give a complete polyhedral description of its convex hull when the multilinear terms in the problem have a nested structure.