The moment-SOS hierarchy and the Christoffel–Darboux kernel

Jean B Lasserre

doi:10.1007/s11590-021-01713-4

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).

Highlights

Consider the Polynomial Optimization Problem (POP): f ∗ = min{ f (x) : x ∈ B }, x where B ⊂ Rn is a compact basic semi-algebraic set
To define an SOS-hierarchy of upper bounds converging to the global minimum f ∗ as described in e.g. [1, 4, 9], we use a measure μ whose support is exactly B, and for which all moments μα := xα dμ, α ∈ Nn, B
To approximate f ∗ from below, consider the hierarchy of semidefinite programs indexed by t ∈ N: m

Summary

Introduction

In (1.3) one searches for an SOS density ( a positive density) whereas in the dual of (1.2) one searches for a signed polynomial density whose coefficients (in the basis of orthonormal polynomials) are moments of a measure on B (ideally the Dirac at a global minimizer). At last but not least, this interpretation establishes another (and rather surprising) simple link between polynomial optimization (here the MomentSOS hierarchy), the Christoffel-Darboux kernel and the Christoffel function, fundamental tools in the theory of orthogonal polynomials and the theory of approximation Previous contributions in this vein include [6] to characterize upper bounds (1.3), [1, 9, 10] to analyze their rate of convergence to f ∗, and the more recent [11] for rate of convergence of both upper and lower bounds on B = {0, 1}n

Main result

Discussion

Conclusion