Variational analysis of pseudospectra

SIAM Journal on Optimization

Volumn 19, Issue 3, 2008, Pages 1048-1072

  Lewis, Adrian S ^a,b   Pang, Jeffrey ^a,b  

^a School of Operations Research and Industrial Engineering   (United States)

^b NONE   (United States)

Author keywords

Aubin property; Lipschitz multifunction; Nonsmooth analysis; Normal cone; Pseudospectrum; Variational analysis

Indexed keywords

AUBIN PROPERTY; LIPSCHITZ MULTIFUNCTION; NONSMOOTH ANALYSIS; NORMAL CONE; PSEUDOSPECTRUM; VARIATIONAL ANALYSIS;

COALESCENCE; EIGENVALUES AND EIGENFUNCTIONS; EQUATIONS OF STATE;

VARIATIONAL TECHNIQUES;

EID: 69649101652     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/070681521     Document Type: Article

References (24)

1. R. Alam and S. Bora, On sensitivity of eigenvalues and eigendecompositions of matrices, Linear Algebra Appl., 396 (2005), pp. 273-301.
2. B. Bank, J. Guddat, D. Klatte, B. Kummer, and K. Tammer, Nonlinear Parametric Optimization, Akademie-Verlag, Berlin, 1982.
3. J. Bolte, A. Daniilidis, A. S. Lewis, and M. Shiota, Clarke critical values of subanalytic Lipschitz continuous functions, Ann. Polon. Math., 87 (2005), pp. 13-25.
4. J. V. Burke, A. S. Lewis, and M. L. Overton, Optimization and pseudospectra, with applications to robust stability, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 80-104. Corrigendum available online from www.cs.nyu.edu/cs/faculty/overton/papers/pseudo-corrigendum.html.
5. J. V. Burke, A. S. Lewis, and M. L. Overton, Robust stability and a criss-cross algorithm for pseudospectra, IMA J. Numer. Anal., 23 (2003), pp. 359-375.
6. J. V. Burke, A. S. Lewis, and M. L. Overton, Convexity and Lipschitz behavior of small pseudospectra, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 586-595.
7. J. V. Burke, A. S. Lewis, and M. L. Overton, Spectral conditioning and pseudospectral growth, Numer. Math., 107 (2007), pp. 27-37.
8. F. Chaitin-Chatelin, A. Harrabi, and A. Ilahi, About Hölder condition numbers and the stratification diagram for defective eigenvalues, Math. Comput. Simulation, 54 (2000), pp. 397-402.
9. F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983. Reprinted by SIAM, Philadelphia, 1990.
10. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.
11. M. Coste, An Introduction to O-Minimal Geometry, http://perso.univ- rennesl.fr/michel. coste/articles.html.
12. J. W. Demmel, A counterexample for two conjectures about stability, IEEE Trans. Automat. Control, 32 (1987), pp. 340-342.
13. M. Embree and L. N. Trefethen, Pseudospectra Gateway, http://web.comlab.ox.ac.uk/projects/pseudospectra/.
14. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
15. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
16. P. Huard, Background to point-to-set maps in mathematical programming, Math. Program. Stud., 10 (1979), pp. 1-28.
17. M. Karow, Geometry of Spectral Value Sets, Ph. D. Thesis, University of Bremen, Bremen, Germany, 2003.
18. M. Karow, Eigenvalue condition numbers and a formula of Burke, Lewis and Overton, Electron. J. Linear Algebra, 15 (2006), pp. 143-153.
19. E. Mengi and M. Overton, Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix, IMA J. Numer. Anal., 25 (2005), pp. 648-669.
20. B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vols I and II, Springer, New York, 2006.
21. R. T. ROCKAFELLAR and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998.
22. L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, Princeton, NJ, 2005.
23. G. A. Watson, Characterization of the subdifferential of some matrix norms, Linear Algebra Appl., 170 (1992), pp. 33-45.
24. T. G. Wright, EigTool: A graphical tool for nonsymmetric eigenproblems, http://www.comlab.ox.ac.uk/pseudospectra/eigtool/.