Global Optimization of Nonconvex Problems with Multilinear Intermediates

We consider global optimization of nonconvex problems containing multilinear functions. It is well known that the convex hull of a multilinear function over a box is polyhedral, and the facets of this polyhedron can be obtained by solving a linear optimization problem (LP). When used as cutting planes, these facets can significantly enhance the quality of conventional relaxations in general-purpose global solvers. However, in general, the size of this LP grows exponentially with the number of variables in the multilinear function. To cope with this growth, we propose a graph decomposition scheme that exploits the structure of a multilinear function to decompose it to lower-dimensional components, for which the aforementioned LP can be solved very efficiently by employing a customized simplex algorithm. We embed this cutting plane generation strategy at every node of the branch-and-reduce global solver BARON, and carry out an extensive computational study on quadratically constrained quadratic problems, multilinear problems, and polynomial optimization problems. Results show that the proposed multilinear cuts enable BARON to solve many more problems to global optimality and lead to an average 60% CPU time reduction.