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The multilinear polytope of a hypergraph is the convex hull of a set of binary points satisfying a collection of multilinear equations. We introduce the running intersection inequalities, a new class of facet-defining inequalities for the multilinear polytope. Accordingly, we define a new polyhedral relaxation of the multilinear polytope, referred to as the running intersection relaxation, and identify conditions under which this relaxation is tight. Namely, we show that for kite-free beta-acyclic hypergraphs, a class that lies between gamma-acyclic and beta-acyclic hypergraphs, the running intersection relaxation coincides with the multilinear polytope and it admits a polynomial size extended formulation. 

We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of the set of binary points z satisfying the system of multilinear equations given above. Recently Del Pia and Khajavirad introduced running intersection inequalities, a family of facet-defining inequalities for the multilinear polytope. In this paper we address the separation problem for this class of inequalities. We first prove that separating flower inequalities, a subclass of running intersection inequalities, is NP-hard. Subsequently, for multilinear polytopes of fixed degree, we devise an efficient polynomial-time algorithm for separating running intersection inequalities and embed the proposed cutting-plane generation scheme at every node of the branch-and-reduce global solver BARON. To evaluate the effectiveness of the proposed method we consider two test sets: randomly generated multilinear and polynomial optimization problems of degree three and four, and computer vision instances from an image restoration problem Results show that running intersection cuts significantly improve the performance of BARON and lead to an average CPU time reduction of 50% for the random test set and of 63% for the image restoration test set.