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 |
Issue Date: | 2003 | Source: | SIAM journal on matrix analysis and applications, 2003, v. 25, no. 3, p. 766-783 | 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. | Keywords: | Symmetric function Spectral function Nonsmooth analysis Semismooth function |
Publisher: | Society for Industrial and Applied Mathematics | Journal: | SIAM journal on matrix analysis and applications | 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 | Size | Format | |
---|---|---|---|---|
Qi_Semismoothness_Spectral_Functions.pdf | 208.77 kB | Adobe PDF | View/Open |
Page views
109
Last Week
0
0
Last month
Citations as of Apr 14, 2024
Downloads
192
Citations as of Apr 14, 2024
SCOPUSTM
Citations
13
Last Week
0
0
Last month
0
0
Citations as of Apr 19, 2024
WEB OF SCIENCETM
Citations
10
Last Week
0
0
Last month
0
0
Citations as of Apr 18, 2024
Google ScholarTM
Check
Altmetric
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.