Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel--Darboux Kernel

Consider the problem of minimizing a polynomial $f$ over compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates positivity polynomials due Putinar and Schmüdgen. When $\mathbf{X}$ is unit ball or standard simplex, we show that Schmüdgen-type converge global minimum at rate in $O(1/r^2)$, matching recently obtained convergence rates for hypersphere hypercube $[-1,1]^n$. For our proof, establish connection between Lasserre's Christoffel--Darboux kernel, make use closed form expressions kernel derived by Xu.