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# Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems

Chen, Xin, Qi, Houduo and Tseng, Paul (2003) Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems. SIAM Journal on Optimization, 13 (4), 960-985.

Record type: Article

## Abstract

For any function f from $\mathbb R$ to $\mathbb R$, one can define a corresponding function on the space of n × n (block-diagonal) real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as ($\rho$-order) semismoothness. Our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices. We also apply our results to the semidefinite complementarity problem, addressing some basic issues in the analysis of smoothing/semismooth Newton methods for solving this problem

Published date: October 2003
Keywords: symmetric-matrix-valued function, nonsmooth analysis, semismooth function, semidefinite complementarity problem
Organisations: Operational Research

## Identifiers

Local EPrints ID: 29643
URI: http://eprints.soton.ac.uk/id/eprint/29643
ISSN: 1052-6234
ORCID for Houduo Qi: orcid.org/0000-0003-3481-4814

## Catalogue record

Date deposited: 12 May 2006

## Contributors

Author: Xin Chen
Author: Houduo Qi
Author: Paul Tseng