  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.
