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This paper introduces cutting planes that involve minimal structuralassumptions, enabling the generation of strong polyhedral relaxations for abroad 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 thatprovides the distance from a point to $S$, we construct a pure cutting planealgorithm which is shown to converge if the initial relaxation is a polyhedron.These cuts are generated from convex forbidden zones, or $S$-free sets, derivedfrom the oracle. We also consider the special case of polynomial optimization.Accordingly we develop a theory of \emph{outer-product-free} sets, where $S$ isthe set of real, symmetric matrices of the form $xx^T$. All maximalouter-product-free sets of full dimension are shown to be convex cones and weidentify several families of such sets. These families are used to generatestrengthened intersection cuts that can separate any infeasible extreme pointof a linear programming relaxation efficiently. Computational experimentsdemonstrate the promise of our approach.

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