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# Semismoothness of spectral functions

Record type: Article

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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## Citation

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

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

## Catalogue record

Date deposited: 12 May 2006

## Contributors

Author: Houduo Qi
Author: Xiaoqi Qi Yang