Nonsmooth Analysis of Singular Values. Part II: Applications |
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Authors: | Adrian S Lewis and Hristo S Sendov |
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Institution: | (1) Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada;(2) Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada |
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Abstract: | In this work we continue the nonsmooth analysis of absolutely symmetric functions of the singular values of a real rectangular matrix. 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.
Mathematics Subject Classifications (2000) Primary 90C31, 15A18; secondary 49K40, 26B05.Research supported by NSERC. |
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Keywords: | nonsmooth analysis singular values regular subdifferential limiting subdifferential proximal subdifferential Clarke subdifferential lower semicontinuous Lidskii |
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