On decomposability of Multilinear sets

Abstract

We consider the Multilinear set ${\mathcal {S}}$ defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form $y_I = \prod _{i \in I} x_i$, $I \in {\mathcal {I}}$, where ${\mathcal {I}}$ denotes a family of subsets of $\{1,\ldots , n\}$ 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 ${\mathcal {S}}$ is decomposable into simpler Multilinear sets ${\mathcal {S}}_j$, $j \in J$; namely, the convex hull of ${\mathcal {S}}$ can be obtained by convexifying each ${\mathcal {S}}_j$, 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 ${\mathcal {S}}$ is decomposable into ${\mathcal {S}}_j$, $j \in J$, 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.