Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization

Abstract

We consider the problem of minimizing a continuous function f over a compact set $${\mathbf {K}}$$K. 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 $${\mathbf {K}}$$K which is a sum of squares of polynomials, so that the expectation $$\int _{{\mathbf {K}}} f(x)h(x)dx$$?Kf(x)h(x)dx is minimized. We show that the rate of convergence is no worse than $$O(1/\sqrt{r})$$O(1/r), where 2r is the degree bound on the density function. This analysis applies to the case when f is Lipschitz continuous and $${\mathbf {K}}$$K 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 $$2r+d$$2r+d of the Lebesgue measure on $${\mathbf {K}}$$K are known, which holds, for example, if $${\mathbf {K}}$$K is a simplex, hypercube, or a Euclidean ball.