Analysis of nonsmooth vector-valued functions associated with second-order cones

Jein Shan Chen*, Xin Chen, Paul Tseng

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

87 Citations (Scopus)


Let [InlineMediaObject not available: see fulltext.] be the Lorentz/second-order cone in [InlineMediaObject not available: see fulltext.]. For any function f from [InlineMediaObject not available: see fulltext.] to [InlineMediaObject not available: see fulltext.], one can define a corresponding function f soc(x) on [InlineMediaObject not available: see fulltext.] by applying f to the spectral values of the spectral decomposition of x [InlineMediaObject not available: see fulltext.] with respect to [InlineMediaObject not available: see fulltext.]. We show that this vector-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as (ρ-order) semismoothness. These results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second-order cone programs and complementarity problems.

Original languageEnglish
Pages (from-to)95-117
Number of pages23
JournalMathematical Programming
Issue number1
Publication statusPublished - 2004 Sept
Externally publishedYes


  • Complementarity
  • Nonsmooth analysis
  • Second-order cone
  • Semismooth function
  • Vector-valued function

ASJC Scopus subject areas

  • Software
  • General Mathematics


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