Nonconvergence of a sum-of-squares hierarchy for global polynomial optimization based on push-forward measures

Let X ⊆ Rⁿ be a closed set, and consider the problem of computing the minimum fₘᵢₙ of a polynomial f on X. Given a measure μ supported on X, Lasserre (SIAM J. Optim. 21(3), 2011) proposes a decreasing sequence of upper bounds on fₘᵢₙ, each of which may be computed by solving a semidefinite program. When X is compact, these bounds converge to fₘᵢₙ under minor assumptions on μ. 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 μ 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 fₘᵢₙ is guaranteed when X = Rⁿ and μ is a Gaussian distribution, we prove that the bounds relying on the push-forward measure fail to converge to fₘᵢₙ in that setting already for polynomials of degree $6$.