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Optimizing matrix stability
Author(s):
J.
V.
Burke;
A.
S.
Lewis;
M.
L.
Overton
Journal:
Proc. Amer. Math. Soc.
129
(2001),
1635-1642.
MSC (2000):
Primary 15A42, 90C30;
Secondary 65F15, 49K30
Posted:
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.
References:
-
- 1.
- V.I. Arnold.
On matrices depending on parameters.
Russian Mathematical Surveys, 26:29-43, 1971. MR 46:400 - 2.
- J.V. Burke and M.L. Overton.
Variational analysis of non-Lipschitz spectral functions, September 1999.
To appear in Mathematical Programming. - 3.
- J.V. Burke, A.S. Lewis, and M.L. Overton. Optimal stability and eigenvalue multiplicity, June 2000. Submitted to Foundations of Computational Mathematics.
- 4.
- R.T. Rockafellar and R.J.-B. Wets.
Variational analysis.
Springer, Berlin, 1998. MR 98m:49001 - 5.
- L.N. Trefethen.
Pseudospectra of linear operators.
SIAM Review, 39:383-406, 1997. MR 98i:47004
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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:
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
Posted:
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
Copyright of article:
Copyright
2000,
American Mathematical Society
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