The moment-SOS hierarchy, which is based on Putinar-type (quadratic module) and Schmüdgen-type (preordering) sum-of-squares positivity certificates, is a widely applicable framework to address polynomial optimization problems over basic semi-algebraic sets. 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 \(\textrm{O}(1/r^2)\) , where r denotes the level of the moment-SOS hierarchy. This paper aims to provide a comprehensive understanding of the convergence rate of the Schmüdgen-type moment-SOS hierarchy by estimating the Hausdorff distance between the set of truncated pseudo-moment sequences and the set of truncated moment sequences through a projection-based method derived from Tchakaloff’s theorem. Our results provide a connection between the convergence rate of the Schmüdgen-type moment-SOS hierarchy and the Łojasiewicz exponent \(\textit{\L} \) of the domain under the compactness assumption, where we establish the convergence rate of \(\textrm{O}(1/r^\textit{\L} )\) . Consequently, we obtain the convergence rate of \(\textrm{O}(1/r)\) for polytopes and sets satisfying the constraint qualification condition, \(\textrm{O}(1/\sqrt{r})\) for domains that either satisfy the Polyak-Łojasiewicz condition or are defined by locally strongly convex polynomials. We also use our method to reprove the convergence rate of \(\textrm{O}(1/r^2)\) for general polynomials over a sphere.
