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

Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems
Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems
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
symmetric-matrix-valued function, nonsmooth analysis, semismooth function, semidefinite complementarity problem
1052-6234
960-985
Chen, Xin
927cf7ef-386c-42b5-aac4-c35836675619
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Tseng, Paul
42eb9b6c-3856-4af3-9fa4-38c81e9d16fd
Chen, Xin
927cf7ef-386c-42b5-aac4-c35836675619
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Tseng, Paul
42eb9b6c-3856-4af3-9fa4-38c81e9d16fd

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. (doi:10.1137/S1052623400380584).

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

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More information

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
PURE UUID: ab10200a-171b-41cf-a2e0-1bac1010ad90
ORCID for Houduo Qi: ORCID iD orcid.org/0000-0003-3481-4814

Catalogue record

Date deposited: 12 May 2006
Last modified: 09 Jan 2022 03:17

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Contributors

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

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