NONCONVERGENCE OF A SUM-OF-SQUARES HIERARCHY FOR GLOBAL POLYNOMIAL OPTIMIZATION BASED ON PUSH-FORWARD MEASURES

Let X subset of R n be a closed set, and consider the problem of computing the minimum f min of a polynomial f on X. Given a measure mu supported on X, Lasserre (SIAM J. Optim. 21(3), 2011) proposes a decreasing sequence of upper bounds on f min , each of which may be computed by solving a semidefinite program. When X is compact, these bounds converge to f min under minor assumptions on mu. Later, Lasserre (Math. Program. 190, 2020) introduces a related, but far more economical sequence of upper bounds which rely on the push-forward measure of mu by f. While these new bounds are weaker a priori, , they actually achieve similar asymptotic convergence rates on compact sets. In this work, we show that no such free lunch exists in the non-compact setting. While convergence of the standard bounds to f min is guaranteed when X = R n and mu is a Gaussian distribution, we prove that the bounds relying on the push-forward measure fail to converge to f min in that setting already for polynomials of degree 6.