  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.
