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

Etienne de Klerk (1) , Monique Laurent (2)

1 Tilburg University [Netherlands]
2 CWI - Centrum voor Wiskunde en Informatica

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.

Mots clés: polynomial optimization on sphere · Lasserre hierarchy · semidefinite programming · generalized eigenvalue problem AMS subject classification 90C22; 90C26; 90C30

Domaines: Mathématiques [math]; Optimisation et contrôle [math.OC]

Etienne de Klerk, Monique Laurent. Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere. 2019, ⟨10.1007/s10107-019-01465-1⟩. ⟨hal-02407335⟩