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A Convex Optimization Approach to Generalized Moment Problems

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Control and Modeling of Complex Systems

Part of the book series: Trends in Mathematics ((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.

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Author information

Authors and Affiliations

  1. Department of Systems Science and Mathematics Washington University, Washington University, St. Louis, MO, 63130, USA

    Christopher I. Byrnes

  2. Department of Mathematics, Division of Optimization and Systems Theory Royal Institute of Technology, Stockholm, 100 44, Sweden

    Anders Lindquist

Authors
  1. Christopher I. Byrnes
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  2. Anders Lindquist
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Editor information

Editors and Affiliations

  1. Department of Information Physics and Computing, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-8656, Japan

    Koichi Hashimoto

  2. Department of Mathematical Informatics, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-8656, Japan

    Yasuaki Oishi

  3. Department of Applied Analysis and Complex Dynamical Systems, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Yutaka Yamamoto

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Byrnes, C.I., Lindquist, A. (2003). A Convex Optimization Approach to Generalized Moment Problems. In: Hashimoto, K., Oishi, Y., Yamamoto, Y. (eds) Control and Modeling of Complex Systems. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0023-9_1

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  • DOI: https://doi.org/10.1007/978-1-4612-0023-9_1

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6577-1

  • Online ISBN: 978-1-4612-0023-9

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