Sum-of-squares hierarchies for polynomial > optimization and the Christoffel-Darboux kernel

Consider the problem of minimizing a polynomial f over a compact semialgebraic set X⊆Rn. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schmüdgen. When X is the unit ball or the standard simplex, we show that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of f at a rate in O(1/r2), matching recently obtained convergence rates for the hypersphere and hypercube [−1,1]n. For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel--Darboux kernel, and make use of closed form expressions for this kernel derived by Xu.