Harmonic Hierarchies for Polynomial Optimization

TL;DR

This paper presents new hierarchies of polyhedral approximations for nonnegative polynomial cones on spheres, with proven convergence bounds and an optimization-free algorithm for polynomial minimization, supported by computational experiments.

Contribution

Introduces novel polyhedral hierarchies for polynomial cones and a new optimization-free method for polynomial minimization on spheres, with convergence guarantees.

Findings

Hierarchies converge at quantifiable rates.

Algorithm effectively computes lower bounds.

Implementation demonstrates practical viability.

Abstract

We introduce novel polyhedral approximation hierarchies for the cone of nonnegative forms on the unit sphere in $\mathbb{R}^n$ and for its (dual) cone of moments. We prove computable quantitative bounds on the speed of convergence of such hierarchies. We also introduce a novel optimization-free algorithm for building converging sequences of lower bounds for polynomial minimization problems on spheres. Finally some computational results are discussed, showcasing our implementation of these hierarchies in the programming language Julia.