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 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 satis ed, e.g., for polytopes and the Euclidean ball). The rth upper bound in the hierarchy may be computed using semide nite 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.
Citation
Technical report, Tilburg University and CWI Amsterdam, November 2014.