Mauricio Velasco: Harmonic hierarchies for polynomial optimization

Abstract

The cone of nonnegative multivariate forms of a given degree is a convex set of remarkable beauty and usefulness.
In this talk, we will discuss some recent ideas for approximating this set through polyhedra. We call the resulting approximations harmonic hierarchies since they arise naturally from harmonic analysis on spheres (or equivalently from the representation theory of SO(n)). We will describe theoretical results leading to precise estimates for the quality of these approximations and to a novel "optimization-free" algorithm for polynomial optimization. We will also show some initial computational results with our Julia implementation for harmonic hierarchies. These results are joint work with Sergio Cristancho (UniAndes).