The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank

TL;DR

This paper introduces a convergent hierarchy of outer approximations for polynomial optimization over convex bodies, extending the polarization hierarchy to finite-dimensional convex cones, with applications to nonnegative matrix factorization.

Contribution

It generalizes the polarization hierarchy to convex cones, providing a new method for polynomial optimization over convex bodies with proven convergence.

Findings

Hierarchy converges for convex bodies characterized by linear or semidefinite programs.

Numerical results show very tight approximations for nonnegative matrix factorization.

Application to the nested rectangles problem demonstrates practical effectiveness.

Abstract

We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has previously been introduced for the study of polynomial optimization problems over state spaces of $C^*$-algebras, to convex cones in finite dimensions. If the convex bodies can be characterized by linear or semidefinite programs, then the same is true for our hierarchy. Convergence is proven by relating the problem to a certain de Finetti theorem for general probabilistic theories, which are studied as possible generalizations of quantum mechanics. We apply the method to the problem of nonnegative matrix factorization, and in particular to the nested rectangles problem. A numerical implementation of the third level of the hierarchy is shown to give rise to a very tight approximation for this problem.