A Polyhedral Study of Binary Polynomial Programs

We study the polyhedral convex hull of a mixed-integer set 𝒮 defined by a collection of multilinear equations over the unit hypercube. Such sets appear frequently in the factorable reformulation of mixed-integer nonlinear optimization problems. In particular, the set 𝒮 represents the feasible region of a linearized unconstrained binary polynomial optimization problem. We define an equivalent hypergraph representation of the mixed-integer set 𝒮, which enables us to derive several families of facet-defining inequalities, structural properties, and lifting operations for its convex hull in the space of the original variables. Our theoretical developments extend several well-known results from the Boolean quadric polytope and the cut polytope literature, paving a way for devising novel optimization algorithms for nonconvex problems containing multilinear sub-expressions.