We consider the problem of minimizing a continuous function f over a compact set (Formula presented.). We analyze a hierarchy of upper bounds proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), obtained by searching for an optimal probability density function h on (Formula presented.) which is a sum of squares of polynomials, so that the expectation (Formula presented.) is minimized. We show that the rate of convergence is no worse than (Formula presented.), where 2r is the degree bound on the density function. This analysis applies to the case when f is Lipschitz continuous and (Formula presented.) is a full-dimensional compact set satisfying some boundary condition (which is satisfied, e.g., for convex bodies). The rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d, and if all moments of order up to (Formula presented.) of the Lebesgue measure on (Formula presented.) are known, which holds, for example, if (Formula presented.) is a simplex, hypercube, or a Euclidean ball.

Mathematical Programming
Networks and Optimization

de Klerk, E., Laurent, M., & Sun, Z. (2017). Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization. Mathematical Programming, 162(1), 363–392. doi:10.1007/s10107-016-1043-1