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 $f_{\min ,K}$ of a polynomial $f$ over a compact set $K\subseteq {\mathbb{R}}^n$, which can be reformulated as finding a probability measure $\nu$ on $K$ minimizing $\int_K f d\nu$\mathrm{.}Lasserreshowedthatitsufficestoconsidersuchmeasuresoftheform$\nu = q\mu$,where$q$isasum-of-squarespolynomialand$\mu$ 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 $f_{\min ,K}$. When $K$ is the hypercube $[-1,1{]}^n$, equipped with the Chebyshev measure, the parameters $f^{(r)}$ are known to converge to $f_{\min ,K}$ at a rate in $O(1\mathrm{/}{r}^2)$. 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$K$satisfiesaminorgeometricalcondition,andin$O(\log^2 r / r^2)$when$K$ is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in $O(1\mathrm{/}\sqrt{r})$ and $O(1\mathrm{/}r)$ for these two respective cases.