Intersection Cuts for Polynomial Optimization

We consider dynamically generating linear constraints (cutting planes) to tighten relaxations for polynomial optimization problems. Many optimization problems have feasible set of the form \(S \cap P\), where S is a closed set and P is a polyhedron. Integer programs are in this class and one can construct intersection cuts using convex “forbidden” regions, or S-free sets. Here, we observe that polynomial optimization problems can also be represented as a problem with linear objective function over such a feasible set, where S is the set of real, symmetric matrices representable as outer-products of the form \(xx^T\). Accordingly, we study outer-product-free sets and develop a thorough characterization of several (inclusion-wise) maximal intersection cut families. In addition, we present a cutting plane approach that guarantees polynomial-time separation of an extreme point in \(P\setminus S\) using our outer-product-free sets. Computational experiments demonstrate the promise of our approach from the point of view of strength and speed.