Variational analysis of convexly generated spectral max functions

Variational analysis of convexly generated spectral max functions The spectral abscissa is the largest real part of an eigenvalue of a matrix and the spectral radius is the largest modulus. Both are examples of spectral max functions—the maximum of a real-valued function over the spectrum of a matrix. These mappings arise in the control and stabilization of dynamical systems. In 2001, Burke and Overton characterized the regular subdifferential of the spectral abscissa and showed that the spectral abscissa is subdifferentially regular in the sense of Clarke when all active eigenvalues are nonderogatory. In this paper we develop new techniques to obtain these results for the more general class of convexly generated spectral max functions. In particular, we extend the Burke–Overton subdifferential regularity result to this class. These techniques allow us to obtain new variational results for the spectral radius. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Programming Springer Journals

Variational analysis of convexly generated spectral max functions

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2016 by Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics
ISSN
0025-5610
eISSN
1436-4646
D.O.I.
10.1007/s10107-016-1088-1
Publisher site
See Article on Publisher Site

Abstract

The spectral abscissa is the largest real part of an eigenvalue of a matrix and the spectral radius is the largest modulus. Both are examples of spectral max functions—the maximum of a real-valued function over the spectrum of a matrix. These mappings arise in the control and stabilization of dynamical systems. In 2001, Burke and Overton characterized the regular subdifferential of the spectral abscissa and showed that the spectral abscissa is subdifferentially regular in the sense of Clarke when all active eigenvalues are nonderogatory. In this paper we develop new techniques to obtain these results for the more general class of convexly generated spectral max functions. In particular, we extend the Burke–Overton subdifferential regularity result to this class. These techniques allow us to obtain new variational results for the spectral radius.

Journal

Mathematical ProgrammingSpringer Journals

Published: Nov 26, 2016

References

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