Semismoothness of spectral functions
Qi, Houduo and Yang, Xiaoqi Qi (2004) Semismoothness of spectral functions. SIAM Journal on Matrix Analysis and Applications, 25, (3), 784803. (doi:10.1137/S0895479802417921).
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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 symmetricmatrixvalued functions.
Item Type:  Article  

Digital Object Identifier (DOI):  doi:10.1137/S0895479802417921  
ISSNs:  08954798 (print) 

Related URLs:  
Keywords:  symmetric function, spectral function, nonsmooth analysis, semismooth function  
Subjects:  Q Science > QA Mathematics  
Divisions :  University Structure  Pre August 2011 > School of Mathematics > Operational Research 

ePrint ID:  29647  
Accepted Date and Publication Date: 


Date Deposited:  12 May 2006  
Last Modified:  06 Aug 2015 02:30  
URI:  http://eprints.soton.ac.uk/id/eprint/29647 
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