Title: Moments, Sums of Squares and Semidefinite Programming
نویسنده
چکیده
We introduce the generalized problem of moments (GPM) which 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, control, signal processing, chemistry, cristallography, tomography, etc. If in full generality the GPM is unsolvable numerically, the situation is much nicer for the GPM with polynomial data. Indeed, one may then define a hierarchy of semidefinite relaxations that provide lower bounds of increasing quality on its optimal value. Even more, thanks to powerful results of real algebraic geometry, one may prove convergence to the optimal value. We will describe various applications that can be formulated as GPM instances in different areas and introduce a hierarchy of semidefinite relaxations for the general GPM which illustrates the duality between moments and sums of squares. Finally, we show how to exploit powerful results from real agebraic geometry to prove convergence to the optimal value of the GPM.
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تاریخ انتشار 2008