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

Cutting planes are derived from specific problem structures, such as a single linear constraint from an integer program. This paper introduces cuts 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; if the initial relaxation is a polytope, the algorithm is shown to converge. Cuts are generated from convex forbidden zones, or $S$-free sets derived from the oracle. We also consider the special case of polynomial optimization. Polynomial optimization may be represented using a symmetric matrix of variables, and in this lifted representation we can let $S$ be the set of matrices that are real, symmetric outer products. Accordingly we develop a theory of \emph{outer-product-free} sets. All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify two families of such sets. These families can be used to generate intersection cuts that can separate any infeasible extreme point of a linear programming relaxation in polynomial time. Moreover, in the special case of polynomial optimization we derive strengthened oracle-based intersection cuts that can also ensure separation in polynomial time.

## Citation

Manuscript, Columbia University, 04/2017. Submitted to Mathematical Programming A.