The Moment-SOS hierarchy and the Christoffel function

The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on $K$. It turns out that this methodology can be also applied to solve problems with positivity constraints "$f(x) >0$ for all $x$ in $K$" and/or linear constraints on Borel measures. Such problems can be viewed as specific instances of the "Generalized Moment Problem" (GMP) whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and in two other applications as well for illustration purpose. Finally we also introduce the Christoffel function (CF), an old tool from the theory of approximation and orthogonal polynomials. It turns out that a non-standard application of the CF is very useful and efficient to recover optimal functions from moments of a measure supported on its graph. Its link with Moment-SOS hierarchy will be described.