Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets

We consider the problem of computing the minimum value fmin, K of a polynomial f over a compact set K. Rn, which can be reformulated as finding a probability measure. on K minimizing K f d.. Lasserre showed that it suffices to consider such measures of the form. = q mu, where q is a sum-of-squares polynomial and mu is a given Borel measure supported on K. By bounding the degree of q by 2r one gets a converging hierarchy of upper bounds f (r) for fmin, K. When K is the hypercube [-1, 1]n, equipped with the Chebyshev measure, the parameters f (r) are known to converge to fmin, K at a rate in O(1/r 2). We extend this error estimate to a wider class of convex bodies, while also allowing for a broader class of reference measures, including the Lebesgue measure. Our analysis applies to simplices, balls and convex bodies that locally look like a ball. In addition, we show an error estimate in O(log r/r) when K satisfies a minor geometrical condition, and in O(log2 r/r 2) when K is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in O(1/v r) and O(1/r) for these two respective cases.