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

• Published in: Mathematical programming Vol. 193; no. 2; pp. 665 - 685
• Main Authors: de Klerk, Etienne, Laurent, Monique
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2022; Springer; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Convergence; Distribution (Probability theory); Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Polynomials; Probability density functions; Theoretical; Upper bounds; Generalized eigenvalue problem; Semidefinite programming; 90C30; Polynomial optimization on sphere; 90C22; Lasserre hierarchy; 90C26
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-019-01465-1

We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), 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 2 r with respect to the surface measure. We show that the rate of convergence is O ( 1 / r 2 ) and we give a class of polynomials of any positive degree for which this rate is tight. In addition, we explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.