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

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