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Geometry&--44; Moments and Semidefinite Optimization

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Abstract:
Optimization formulations seek to take advantage of structure and information in a particular problem, and investigate how successfully this information constrains the performance measures of interest. In this talk, we apply some recent results of algebraic geometry, to show how the underlying geometry of the problem may be incorporated in a natural way, in a semidefinite optimization formulation. We will discuss two examples in depth, to illustrate this concept. The first, we present a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on both linear and certain nonlinear functionals defined on solutions of linear partial differential equations. We apply the proposed methods to examples of PDEs in one and two dimensions with encouraging results. As a second example, we consider deriving optimal inequalities in probability theory, based on partial knowledge of the moments of the distribution, and of its support. In both cases, the additional ``geometry--constraints'' significantly improve the quality of the bounds. Furthermore, we believe this framework to be a general one, with many potential applications.

Type of Seminar:
Public Seminar
Speaker:
Constantine Caramanis
Ph.D. student at M.I.T. in the Laboratory for Information and Decision Systems
Date/Time:
Jun 24, 2002   14:15
Location:

ETH Zentrum, Physikstrasse 3, Building ETL , Room K 25
Contact Person:

Prof. Pablo Parrilo
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Biographical Sketch:
Constantine Caramanis is currently a Ph.D. student at M.I.T. in the Laboratory for Information and Decision Systems, working with Prof. Bertsimas. He received the A.B. degree in Mathematics from Harvard University in 1999.