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Eigenvalue perturbations and nonlinear parametric optimization

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Book cover Nonlinear Analysis and Optimization

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 30))

Abstract

It is well known that the eigenvalues resp. eigenvectors of symmetric matrices can be characterized as optimal values resp. solutions of nonlinear optimization problems. As a novelty, in this paper also the study of parameterized eigenvalue problems is traced back to that of parameterized nonlinear programming problems. We treat families of generalized eigenvalue problems indexed by a (vector-valued) parameter p and study the eigenvalues as functions of p. This is done by employing recent results about the optimal value functions of nonlinear optimization problems. Thus some classical eigenvalue perturbation results are obtained but also new ones are derived, for instance when the data are only Lipschitzian or when p is not scalar.

Throughout the paper emphasis is laid upon the optimization aspects of the problems. Further extensions of the theory employ deeper results from linear algebra and will be reported in a subsequent paper.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Würzburg, Am Hubland, D-8700, Würzburg, FR Germany

    Bernhard Gollan

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  1. Bernhard Gollan
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B. Cornet V. H. Nguyen J. P. Vial

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© 1987 The Mathematical Programming Society, Inc.

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Gollan, B. (1987). Eigenvalue perturbations and nonlinear parametric optimization. In: Cornet, B., Nguyen, V.H., Vial, J.P. (eds) Nonlinear Analysis and Optimization. Mathematical Programming Studies, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121155

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  • DOI: https://doi.org/10.1007/BFb0121155

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00930-3

  • Online ISBN: 978-3-642-00931-0

  • eBook Packages: Springer Book Archive

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