On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences

Abstract: The moment-SOS hierarchy is a widely applicable framework to address polynomial optimization problems over basic semi-algebraic sets based on positivity certificates of polynomial. Recent works show that the convergence rate of this hierarchy over certain simple sets, namely, the unit ball, hypercube, and standard simplex, is of the order $O(1/r^2)$, where r denotes the level of the moment-SOS hierarchy. This paper aims to provide a comprehensive understanding of the convergence rate of the moment-SOS hierarchy by estimating the Hausdorff distance between the set of truncated pseudo-moment sequences and the set of truncated moment sequences specified by Tchakaloff's theorem. Our results provide a connection between the convergence rate of the moment-SOS hierarchy and the Lojasiewicz exponent L of the domain under the compactness assumption, where we establish the convergence rate of $O(1/r^L)$. Consequently, we obtain the convergence rate of $O(1/r)$ for polytopes and sets satisfying the constraint qualification condition, $O(1/\sqrt{r})$ for domains that either satisfy the Polyak-Lojasiewicz condition or are defined by locally strongly convex polynomials. We also obtain the convergence rate of $O(1/r^2)$ for general polynomials over a sphere.