Harmonic Hierarchies for Polynomial Optimization

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.