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

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11 Citations (Scopus)

Abstract

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
Volume85
DOIs
Publication statusPublished - 2013 Apr 3

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Circular cone
Vector-valued Functions
Differentiability
Cones
Directional Differentiability
Semismoothness
Second-order Cone
Lipschitz Continuity
Spectral Decomposition
Optimization Problem
Decomposition

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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title = "Smooth and nonsmooth analyses of vector-valued functions associated with circular cones",
abstract = "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 Lθ. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fr{\'e}chet 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.",
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author = "Yu-Lin Chang and Yang, {Ching Yu} and Jein-Shan Chen",
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AU - Chang, Yu-Lin

AU - Yang, Ching Yu

AU - Chen, Jein-Shan

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N2 - 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 Lθ. 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.

AB - 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 Lθ. 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.

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