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.

最小化二元变量的多线性函数的问题是一个经过充分研究的 NP 难题。该问题的标准线性化的解集称为多线性集。我们研究了它的基数约束版本，其中非零变量的数量有上限和下限。我们将该问题的标准线性化的解集称为具有基数约束的多线性集。我们在这些多线性项上描述一组条件（称为适当性），并观察到在这些条件下凸包该集合的描述可以通过扩展公式来处理。然后，当多线性项具有嵌套结构时，我们给出凸包的显式多面体描述。我们的描述有指数数量的不等式，可以在多项式时间内分离。最后，我们概括这些不等式以获得一般情况下的有效不等式。