The moment-SOS hierarchy and the Christoffel–Darboux kernel

Abstract

We consider the global minimization of a polynomial on a compact set $\mathbf {B}$. We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of $L^2(\mathbf {B},\mu )$ where $\mu$ is an arbitrary reference measure whose support is exactly $\mathbf {B}$. The resulting polynomial is a certain density (with respect to $\mu$) of some signed measure on $\mathbf {B}$. When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel–Darboux (CD) kernel and the Christoffel function associated with $\mu$. In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).