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Semismoothness of spectral functions

Semismoothness of spectral functions
Semismoothness of spectral functions
Any spectral function can be written as a composition of a symmetric function $f: \rn \mapsto \Re$ and the eigenvalue function $\lambda(\cdot): \s \mapsto \rn$, often denoted by $(f \circ \lambda)$, where $\s$ is the subspace of n × n symmetric matrices. In this paper, we present some nonsmooth analysis for such spectral functions. Our main results are (a) $(f \circ \lambda)$ is directionally differentiable if f is semidifferentiable, (b) $(f \circ \lambda)$ is LC 1 if and only if f is LC 1, and (c) $(f \circ \lambda)$ is SC 1 if and only if f is SC 1. Result (a) is complementary to a known (negative) fact that $(f \circ \lambda)$ might not be directionally differentiable if f is directionally differentiable only. Results (b) and (c) are particularly useful for the solution of LC 1 and SC 1 minimization problems which often can be solved by fast (generalized) Newton methods. Our analysis makes use of recent results on continuously differentiable spectral functions as well as on nonsmooth symmetric--matrix-valued functions.
symmetric function, spectral function, nonsmooth analysis, semismooth function
0895-4798
784-803
Qi, Houduo
864c4f3b-2a55-45b5-842b-c52f2de0565c
Yang, Xiaoqi Qi
168ad561-796e-4575-9f33-47d33d06487f
Qi, Houduo
864c4f3b-2a55-45b5-842b-c52f2de0565c
Yang, Xiaoqi Qi
168ad561-796e-4575-9f33-47d33d06487f

Qi, Houduo and Yang, Xiaoqi Qi (2004) Semismoothness of spectral functions. SIAM Journal on Matrix Analysis and Applications, 25 (3), 784-803. (doi:10.1137/S0895479802417921).

Record type: Article

Abstract

Any spectral function can be written as a composition of a symmetric function $f: \rn \mapsto \Re$ and the eigenvalue function $\lambda(\cdot): \s \mapsto \rn$, often denoted by $(f \circ \lambda)$, where $\s$ is the subspace of n × n symmetric matrices. In this paper, we present some nonsmooth analysis for such spectral functions. Our main results are (a) $(f \circ \lambda)$ is directionally differentiable if f is semidifferentiable, (b) $(f \circ \lambda)$ is LC 1 if and only if f is LC 1, and (c) $(f \circ \lambda)$ is SC 1 if and only if f is SC 1. Result (a) is complementary to a known (negative) fact that $(f \circ \lambda)$ might not be directionally differentiable if f is directionally differentiable only. Results (b) and (c) are particularly useful for the solution of LC 1 and SC 1 minimization problems which often can be solved by fast (generalized) Newton methods. Our analysis makes use of recent results on continuously differentiable spectral functions as well as on nonsmooth symmetric--matrix-valued functions.

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More information

Published date: 2004
Keywords: symmetric function, spectral function, nonsmooth analysis, semismooth function
Organisations: Operational Research

Identifiers

Local EPrints ID: 29647
URI: http://eprints.soton.ac.uk/id/eprint/29647
ISSN: 0895-4798
PURE UUID: 96fcb4e5-30d0-4db5-bb08-d793c289f2ad

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Date deposited: 12 May 2006
Last modified: 15 Mar 2024 07:33

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Contributors

Author: Houduo Qi
Author: Xiaoqi Qi Yang

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