Abstract
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 in {mathbb {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 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.
Highlights
We consider the problem of minimizing an n-variate polynomial f : Rn → R over a compact set K ⊆ Rn, i.e., the problem of computing the parameter: fmin,K
Theorem 11 (De Klerk-Postek-Kuhn [12]) Assume that f0, . . . , fm are polynomials, K is compact, μ is a Borel measure supported on K, and the generalized problem of moments (GPM) (19) has an optimal solution
In view of the fact that we could show the improved O(1/r 2) rate for the upper bounds, and the fact that the lower bounds hierarchy empirically converges much faster in practice, one would expect that the lower bounds (9) converge at a rate no worse than O(1/r 2)
Summary
Our main contribution in this paper is to show that the convergence rate of the bounds f (r) is O(1/r 2) for any polynomial f and, that this analysis is tight for any (nonzero) linear polynomial f (and some powers). This is summarized in the following theorem, where we use the usual Landau notation: for two functions f1, f2 : N → R+, f1.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
