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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Optimizing matrix stability


Authors: J. V. Burke, A. S. Lewis and M. L. Overton
Journal: Proc. Amer. Math. Soc. 129 (2001), 1635-1642
MSC (2000): Primary 15A42, 90C30; Secondary 65F15, 49K30
Published electronically: October 31, 2000
MathSciNet review: 1814091
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Abstract: Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matrices with nontrivial Jordan structure are rare in the space of all matrices, they appear naturally in spectral abscissa minimization.


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Additional Information

J. V. Burke
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: burke@math.washington.edu

A. S. Lewis
Affiliation: Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: aslewis@math.uwaterloo.ca

M. L. Overton
Affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
Email: overton@cs.nyu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05985-2
PII: S 0002-9939(00)05985-2
Keywords: Eigenvalue optimization, spectral abscissa, nonsmooth analysis, Jordan form
Received by editor(s): September 28, 1999
Published electronically: October 31, 2000
Additional Notes: The first author’s research was supported by the National Science Foundation grant number DMS-9971852
The second author’s research was supported by the Natural Sciences and Engineering Research Council of Canada
The third author’s research was supported by the National Science Foundation grant number CCR-9731777, and the U.S. Department of Energy Contract DE-FG02-98ER25352
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2000 American Mathematical Society