Outer-Product-Free Sets for Polynomial Optimization and Oracle-Based Cuts

Abstract

This paper introduces cutting planes that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set $S\cap P$, where $S$ is a closed set, and $P$ is a polyhedron. Given an oracle that provides the distance from a point to $S$, we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron. These cuts are generated from convex forbidden zones, or $S$-free sets, derived from the oracle. We also consider the special case of polynomial optimization. Accordingly we develop a theory of \emph{outer-product-free} sets, where $S$ is the set of real, symmetric matrices of the form $xx^T$. All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify several families of such sets. These families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently. Computational experiments demonstrate the promise of our approach.