Smooth and nonsmooth analyses of vector-valued functions associated with circular cones

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Let Lθ be the circular cone in Rn which includes a second-order cone as a special case. For any function f from R to R, one can define a corresponding vector-valued function fc(x) on Rn by applying f to the spectral values of the spectral decomposition of x ⋯ Rn with respect to . We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as semismoothness. These results will play a crucial role in designing solution methods for optimization problem associated with the circular cone.

Original languageEnglish
Pages (from-to)160-173
Number of pages14
JournalNonlinear Analysis, Theory, Methods and Applications
Publication statusPublished - 2013 Apr 3



  • Circular cone Vector-valued function Semismooth function Complementarity Spectral decomposition

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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