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Lewis, Adrian S.; Sendov, Hristo S.

Nonsmooth analysis of singular values. I: Theory. (English) Zbl 1129.49025

Set-Valued Anal. 13, No. 3, 213-241 (2005).
Summary: The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of functions of singular values. In particular we give simple formulae for the regular subdifferential, the limiting subdifferential, and the horizon subdifferential, of such functions. Along the way to the main result we give several applications and in particular derive von Neumann’s trace inequality for singular values. For Part II see the review below.

Cited in 1 Review
Cited in 42 Documents

MSC:

49J52 Nonsmooth analysis
90C31 Sensitivity, stability, parametric optimization

Keywords:

nonsmooth analysis; singular values; regular subdifferential; limiting; subdifferential; horizon subdifferential; von Neumann trace inequality; simultaneous diagonalization

Citations:

Zbl 1129.49026
Cite Review PDF
Full Text: DOI

References:

[1] Brickell, F. and Clark, R. S.: Differential Manifolds; An Introduction, Van Nostrand Reinhold, London, 1970. · Zbl 0199.56303
[2] Clarke, F. H.: Necessary conditions for nonsmooth problems in optimal control and the calculus of variations, PhD Thesis, University of Washington, 1973.
[3] Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983. · Zbl 0582.49001
[4] Clarke, F. H., Ledyaev, Yu. S., Stern, R. J. and Wolenski, P. R.: Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998. · Zbl 1047.49500
[5] Gibson, P. M.: Simultaneous diagonalization of rectangular complex matrices, Linear Algebra Appl. 9 (1974), 45–53. · doi:10.1016/0024-3795(74)90025-1
[6] Hiriart-Urruty, J.-B. and Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals, Grundlehren Math. Wiss. 305, Springer-Verlag, Berlin, 1993.
[7] Horn, R. A. and Johnson, C. R.: Matrix Analysis, 2nd edn, Cambridge University Press, 1985. · Zbl 0576.15001
[8] Horn, R. A. and Johnson, C. R.: Topics in Matrix Analysis, Cambridge University Press, 1991. Paperback edition with corrections, 1994. · Zbl 0729.15001
[9] Lewis, A. S.: The convex analysis of unitarily invariant matrix functions, J. Convex Anal. 2(1–2) (1994), 173–183. · Zbl 0860.15026
[10] Lewis, A. S.: Convex analysis on the Hermitian matrices, SIAM J. Optim. 6 (1996), 164–177. · Zbl 0849.15013 · doi:10.1137/0806009
[11] Lewis, A. S.: Derivatives of spectral functions, Math. Oper. Res. 21 (1996), 576–588. · Zbl 0860.49017 · doi:10.1287/moor.21.3.576
[12] Lewis, A. S.: Group invariance and convex matrix analysis, SIAM J. Matrix Anal. 17(4) (1996), 927–949. · Zbl 0876.15016 · doi:10.1137/S0895479895283173
[13] Lewis, A. S.: Nonsmooth analysis of eigenvalues, Math. Programming 84 (1999), 1–24. · Zbl 0969.49006
[14] Lewis, A. S.: Convex analysis on Cartan subspaces, Nonlinear Anal. 42 (2000), 813–820. · Zbl 1159.15303 · doi:10.1016/S0362-546X(99)00126-1
[15] Lewis, A. S. and Sendov, H. S.: Nonsmooth analysis of singular values Part II: Applications, Set Valued Anal. 13 (2005), 243–264. · Zbl 1129.49026 · doi:10.1007/s11228-004-7198-6
[16] Rockafellar, R. T. and Wets, R. J.-B.: Variational Analysis, Springer-Verlag, Berlin, 1998.
[17] Sendov, H. S.: Variational spectral analysis, Ph.D. Thesis, University of Waterloo, 2000, http://etd.uwaterloo.ca/etd/hssendov2000.pdf. · Zbl 0960.65146
[18] Tam, T.-Y. and Hill, W. C.: Derivatives of orbital functions, an extension of Berezin–Gel’fand’s theorem and applications, Preprint, http://web6.duc.auburn.edu/\(\sim\)tamtiny/gb2.pdf.
[19] Tam, T. Y.: An extension of a result of Lewis, Electron. J. Linear Algebra 5 (1999), 1–10. · Zbl 0928.15012
[20] Thompson, R. C.: On Pearl’s paper ’A decomposition theorem for matrices’, Canad. Math. Bull. 12(6) (1969), 805–808. · Zbl 0203.33302 · doi:10.4153/CMB-1969-104-5
[21] von Neumann, J.: Some matrix inequalities and metrization of matric-space, Tomsk University Review 1 (1937), 286–300. In: Collected Works, Vol. IV, Pergamon, Oxford, 1962, pp. 205–218. · JFM 63.0037.03
[22] Wiegmann, S.: Some analogues of the generalized principle axis transformation, Bull. Amer. Math. Soc. 54 (1948), 905–908. · Zbl 0032.10301 · doi:10.1090/S0002-9904-1948-09095-4
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