Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

This work has been supported by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement 813211 (POEMA)We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems , proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864 − 885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r ∈ N of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the exact rate of convergence is Θ(1/r 2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere