Course 1: Moments, sums of squares and semidefinite programming.

(Jean-Bernard LASSERRE, Directeur de Recherche CNRS, Francia)

Short description of the course. Roughly speaking, the Generalized Problem of Moments (GPM) is an infinite-dimensional linear optimization problem (i.e., an infinite dimensional linear program) on a convex set of measures with support on a given subset K of Rⁿ. From a theoretical viewpoint, the GPM has developments and impact in various area of Mathematics like Algebra, Fourier Analysis, Functional Analysis, Operator Theory,  Probability and Statistics, to cite a few. In addition, and despite its rather simple and short formulation, the GPM has a large number of important applications in various fields like optimization, probability, mathematical finance, optimal control, control and signal processing, chemistry, crystallography, tomography, quantum computing, etc.

In its full generality, the GPM is intractable numerically. However when the set K is compact and semi-algebraic, and the functions involved are polynomials (and in some cases piecewise polynomials or rational functions), then the situation is much nicer. Indeed, one can define a systematic numerical scheme based on a hierarchy of semidefinite programs, which provides a monotone sequence that converges to the optimal value of the GPM. Sometimes finite convergence may even occur. In the talk, we will present a methodology based on semidefinite programming to solve the GPM and we will describe in detail several applications of the GPM (notably in optimization, probability, optimal control and mathematical finance).