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Nonsmooth analysis of singular values. II: Applications. (English) Zbl 1129.49026
Summary: In this work we continue the nonsmooth analysis of absolutely symmetric functions of the singular values of a real rectangular matrix [see Part I reviewed above]. Absolutely symmetric functions are invariant under permutations and sign changes of its arguments. We extend previous work on subgradients to analogous formulae for the proximal subdifferential and Clarke subdifferential when the function is either locally Lipschitz or just lower semicontinuous. We illustrate the results by calculating the various subdifferentials of individual singular values. Another application gives a nonsmooth proof of Lidskii’s theorem for weak majorization.

49J52Nonsmooth analysis (other weak concepts of optimality)
90C31Sensitivity, stability, parametric optimization
[1]Borwein, J. M. and Lewis, A. S.: Convex Analysis and Nonlinear Optimization, Springer, New York, 2000.
[2]Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
[3]Horn, R. A. and Johnson, C. R.: Matrix Analysis, 2nd edn, Cambridge University Press, 1985.
[4]Horn, R. A. and Johnson, C. R.: Topics in Matrix Analysis, Cambridge University Press, 1991. Paperback edition with corrections, 1994.
[5]Ioffe, A. D.: Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 389–416.
[6]Lewis, A. S.: Convex analysis on the Hermitian matrices, SIAM J. Optim. 6 (1996), 164–177. · Zbl 0849.15013 · doi:10.1137/0806009
[7]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
[8]Lewis, A. S.: Lidskii’s theorem via nonsmooth analysis, SIAM J. Matrix Anal. Appl. 21 (1999), 379–381. · Zbl 1047.90511 · doi:10.1137/S0895479898338676
[9]Lewis, A. S.: Nonsmooth analysis of eigenvalues, Math. Programming 84 (1999), 1–24.
[10]Lewis, A. S. and Sendov, H. S.: Nonsmooth analysis of singular values, Part I: Theory, Set-Valued Anal. (2005), 213–241.
[11]Markus, A. S.: The eigen- and singular values of the sum and product of linear operators, Uspekhi Mat. Nauk 19(4) (1964), 93–123. Russian Math. Surveys 19 (1964), 92–120.
[12]Mordukhovich, B. S.: Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988.
[13]Rockafellar, R. T.: Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
[14]Rockafellar, R. T. and Wets, R. J.-B.: Variational Analysis, Springer-Verlag, Berlin, 1998.
[15]Sendov, H. S.: Variational Spectral Analysis, University of Waterloo, PhD Thesis, 2000, http://etd.uwaterloo.ca/etd/hssendov2000.pdf.
[16]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/tamtiny/gb2.pdf.
[17]Torki, M.: Second-order directional derivatives of all eigenvalues of a symmetric matrix, Nonlinear Anal., Ser. A Theory, Methods 46(8) (2001).
[18]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.