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We consider the problem of minimizing a continuous function f over a compactset K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J.Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimalprobability density function h on K which is a sum of squares of polynomials,so that the expectation $\int_{K}f(x)h(x)dx$ is minimized. We show that therate of convergence is no worse than $O(1/\sqrt{r})$, where 2r is the degreebound on the density function. This analysis applies to the case when f isLipschitz continuous and K is a full-dimensional compact set satisfying someboundary condition (which is satisfied, e.g., for convex bodies). The r-thupper bound in the hierarchy may be computed using semidefinite programming iff is a polynomial of degree d, and if all moments of order up to 2r+d of theLebesgue measure on K are known, which holds for example if K is a simplex,hypercube, or a Euclidean ball.

Convergence analysis for Lasserre's measure--based hierarchy of upper bounds for polynomial optimization