基数约束下的凸化多线性集：结构性质、套情况及推广
Convexifying Multilinear Sets with Cardinality Constraints: Structural Properties, Nested Case and Extensions
摘要：
二元多线性函数的极小化问题是一个NP难问题。这个问题的标准线性化的解的集合被称为多重线性集合。我们研究了一个基数约束的版本，它的上界和下界上的非零变量的数量。我们称这个问题的标准线性化的解的集合为具有基数约束的多线性集合。我们刻画了一组条件，这些多线性项（称为propriness），并观察到，在这些条件下的凸包描述的集合是通过一个扩展的配方易于处理。当多重线性项具有嵌套结构时，我们给出了该凸包的一个显式多面体描述。我们的描述有一个指数数量的不等式，可以在多项式时间内分离。最后，我们推广这些不等式，得到有效的不等式的一般情况。
英文摘要：
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.
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