On Decomposability of Multilinear Sets

We consider the Multilinear set S defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form yI = ∏ i∈I xi, I ∈ I, where I denotes a family of subsets of {1, . . . , n} of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is possible when S is decomposable into simpler Multilinear sets Sj , j ∈ J ; namely, the convex hull of S can be obtained by convexifying each Sj , separately. In this paper, we study the decomposability properties of Multilinear sets. Utilizing an equivalent hypergraph representation for Multilinear sets, we derive necessary and sufficient conditions under which S is decomposable into Sj , j ∈ J , based on the structure of pair-wise intersection hypergraphs. Our characterizations unify and extend the existing decomposability results for the Boolean quadric polytope. Finally, we propose a polynomial-time algorithm to optimally decompose a Multilinear set into simpler subsets. Our proposed algorithm can be easily incorporated in branch-and-cut based global solvers as a preprocessing step for cut generation.