# Semismoothness of spectral functions

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

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## Description/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.

Item Type: Article 0895-4798 (print) http://dx.doi.org/10.1137/S089...9802417921 symmetric function, spectral function, nonsmooth analysis, semismooth function Q Science > QA Mathematics University Structure - Pre August 2011 > School of Mathematics > Operational Research 29647 12 May 2006 01 Jun 2011 06:35 hdqi@soton.ac.uk http://eprints.soton.ac.uk/id/eprint/29647

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