Semismoothness, of spectral functions

Houduo Qi, Xiaoqi Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

13 Citations (Scopus)


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 ο 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 ο λ) is directionally differentiate if f is semidifferentiable, (b) (f ο λ) is LC1 if and only if f is LC1, and (c) (f ο λ) is SC1 if and only if f is SC1. Result (a) is complementary to a known (negative) fact that (f ο 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.
Original languageEnglish
Pages (from-to)766-783
Number of pages18
JournalSIAM Journal on Matrix Analysis and Applications
Issue number3
Publication statusPublished - 26 Jul 2004


  • Nonsmooth analysis
  • Semismooth function
  • Spectral function
  • Symmetric function

ASJC Scopus subject areas

  • Analysis


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