Moments, Positive Polynomials and Their Applications

. From a theoretical viewpoint, the GPM has developments and impact in var-ious area of Mathematics like algebra, Fourier analysis, functional analysis, operator theory, probabilityand statistics, to cite a few. In addition, and despite its rather simple and short formulation, the GPMhas a large number of important applications in various fields like optimization, probability, mathematicalfinance, optimal control, control and signal processing, chemistry, cristallography, tomography, quantumcomputing, etc.In its full generality, the GPM is untractable numerically. However when K is a compact basic semi-algebraic set, and the functions involved are polynomials (and in some cases piecewise polynomials orrational functions), then the situation is much nicer. Indeed, one can define a systematic numerical schemebased on a hierarchy of semidefinite programs, which provides a monotone sequence that converges tothe optimal value of the GPM. (A semidefinite program is a convex optimization problem which up toarbitrary fixed precision, can be solved in polynomial time.) Sometimes finite convergence may evenocccur.In the talk, we will present the semidefinite programming methodology to solve the GPM and describein detail several applications of the GPM (notably in optimization, probability, optimal control andmathematical finance).R´ef´erences[1] J.B. Lasserre, Moments, Positive Polynomials and their Applications, Imperial College Press, inpress.[2] J.B. Lasserre, A Semidefinite programming approach to the generalized problem of moments,Math. Prog. 112 (2008), pp. 65–92.