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A Convex Optimization Approach to Generalized Moment Problems

  • Christopher I. Byrnes
  • Anders Lindquist
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

In this paper we present a universal solution to the generalized moment problem, with a nonclassical complexity constraint. We show that this solution can be obtained by minimizing a strictly convex nonlinear functional. This optimization problem is derived in two different ways. We first derive this intrinsically, in a geometric way, by path integration of a one-form which defines the generalized moment problem. It is observed that this one-form is closed and defined on a convex set, and thus exact with, perhaps surprisingly, a strictly convex primitive function. We also derive this convex functional as the dual problem of a problem to maximize a cross entropy functional. In particular, these approaches give a constructive parameterization of all solutions to the Nevanlinna-Pick interpolation problem, with possible higher-order interpolation at certain points in the complex plane, with a degree constraint as well as all solutions to the rational covariance extension problem — two areas which have been advanced by the work of Hidenori Kimura. Illustrations of these results in system identification and probability are also mentioned.

Keywords

Moment problems Convex optimization Nevanlinna-Pick interpolation Covariance extension Systems identification Kullback-Leibler distance 

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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Christopher I. Byrnes
    • 1
  • Anders Lindquist
    • 2
  1. 1.Department of Systems Science and Mathematics Washington UniversityWashington UniversitySt. LouisUSA
  2. 2.Department of MathematicsDivision of Optimization and Systems Theory Royal Institute of TechnologyStockholmSweden

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