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

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