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

Abstract

AbstractWe consider the problem of computing the minimum value $$f_{\min ,K}$$fmin,K of a polynomial f over a compact set $$K\subseteq {\mathbb {R}}^n$$K⊆Rn, which can be reformulated as finding a probability measure $$\nu $$ν on $$K$$K minimizing $$\int _Kf d\nu $$∫Kfdν. Lasserre showed that it suffices to consider such measures of the form $$\nu = q\mu $$ν=qμ, where q is a sum-of-squares polynomial and $$\mu $$μ is a given Borel measure supported on $$K$$K. By bounding the degree of q by 2r one gets a converging hierarchy of upper bounds $$f^{(r)}$$f(r)  for $$f_{\min ,K}$$fmin,K. When K is the hypercube $$[-1, 1]^n$$[-1,1]n, equipped with the Chebyshev measure, the parameters $$f^{(r)}$$f(r) are known to converge to $$f_{\min , K}$$fmin,K at a rate in $$O(1/r^2)$$O(1/r2). 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 $$O(\log r / r)$$O(logr/r) when $$K$$K satisfies a minor geometrical condition, and in $$O(\log ^2 r / r^2)$$O(log2r/r2) when $$K$$K is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in $$O(1 / \sqrt{r})$$O(1/r) and $$O(1/r)$$O(1/r) for these two respective cases.