Polynomial Optimization, Certificates of Positivity, and Christoffel Function
Abstract
We briefly recall basics of the Moment-SOS hierarchy in polynomial optimization and the Christoffel-Darboux kernel (and the Christoffel function (CF)) in theory of approximation and orthogonal polynomials. We then (i) show a strong link between the CF and the SOS-based positive certificate at the core of the Moment-SOS hierarchy, and (ii) describe how the CD-kernel provides a simple interpretation of the SOS-hierarchy of lower bounds as searching for some signed polynomial density (while the SOS-hierarchy of upper bounds is searching for a positive (SOS) density). This link between the CF and positive certificates, in turn allows us (i) to establish a disintegration property of the CF much like for measures, and (ii) for certain sets, to relate the CF of their equilibrium measure with a certificate of positivity on the set, for constant polynomials.
Cite
@article{arxiv.2304.12772,
title = {Polynomial Optimization, Certificates of Positivity, and Christoffel Function},
author = {Jean-Bernard Lasserre},
journal= {arXiv preprint arXiv:2304.12772},
year = {2023}
}