Moments, Positive Polynomials and Semidefinite Programming

Abstract: 

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 whose support is on a given subset K. 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, cristallography, tomography, quantum computing, etc.

In full generality, the GPM is untractable numerically. However when the set K is compact and semi-algebraic, and the functions involved are polynomials (and even semi-algebraic) then the situation is much nicer. Indeed, invoking powerful (representation) results of real algebraic geometry, one can define a systematic numerical scheme (the moment-SOS approach) based on a hierarchy of semidefinite programs of increasing size. The associated sequence of optimal values is monotone and converges to the optimal value of the GPM. In practice finite converge takes place very often and is even guaranteed in the discrete case.

In the talk, we will describe the moment-SOS approach to solve the GPM and describe in detail several applications of the GPM (notably in optimization, probability, optimal control and mathematical finance).