Worst-case examples for Lasserre's measure--based hierarchy for polynomial optimization on the hypercube

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim. 27(1), (2017) pp. 347-367]. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials.