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arXiv · 2304.12772

Polynomial Optimization, Certificates of Positivity, and Christoffel Function

Jean-Bernard Lasserre
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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.

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Jean-Bernard Lasserre. 2023-04-25. Polynomial Optimization, Certificates of Positivity, and Christoffel Function. https://arxiv.org/abs/2304.12772

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