Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/6105
Title: Semismoothness of spectral functions
Authors: Qi, HD
Yang, XQ 
Keywords: Symmetric function
Spectral function
Nonsmooth analysis
Semismooth function
Issue Date: 2003
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on matrix analysis and applications, 2003, v. 25, no. 3, p. 766-783 How to cite?
Journal: SIAM journal on matrix analysis and applications 
Abstract: Any spectral function can be written as a composition of a symmetric function f : ℝⁿ ↦ ℝ and the eigenvalue function λ(·): S ↦ℝⁿ, often denoted by (f◦λ), 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◦λ) is directionally differentiable if f is semidifferentiable, (b) (f◦λ) is LC¹ if and only if f is LC¹ , and (c) (f◦λ) is SC¹ if and only if f is SC¹ . Result (a) is complementary to a known (negative) fact that (f◦λ) might not be directionally differentiable if f is directionally differentiable only. Results (b) and (c) are particularly useful for the solution of LC¹ and SC¹ 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.
URI: http://hdl.handle.net/10397/6105
ISSN: 0895-4798
EISSN: 1095-7162
DOI: 10.1137/S0895479802417921
Rights: © 2004 Society for Industrial and Applied Mathematics
Appears in Collections:Journal/Magazine Article

Files in This Item:
File Description SizeFormat 
Qi_Semismoothness_Spectral_Functions.pdf208.77 kBAdobe PDFView/Open
Access
View full-text via PolyU eLinks SFX Query
Show full item record

SCOPUSTM   
Citations

10
Last Week
0
Last month
0
Citations as of Aug 15, 2017

Page view(s)

101
Last Week
0
Last month
Checked on Aug 20, 2017

Download(s)

81
Checked on Aug 20, 2017

Google ScholarTM

Check

Altmetric



Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.