  We consider the problem of minimizing a continuous function f over a compact
set 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 optimal
probability 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 the
rate of convergence is no worse than $O(1/\sqrt{r})$, where 2r is the degree
bound on the density function. This analysis applies to the case when f is
Lipschitz continuous and K is a full-dimensional compact set satisfying some
boundary condition (which is satisfied, e.g., for convex bodies). The r-th
upper bound in the hierarchy may be computed using semidefinite programming if
f is a polynomial of degree d, and if all moments of order up to 2r+d of the
Lebesgue measure on K are known, which holds for example if K is a simplex,
hypercube, or a Euclidean ball.
