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

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1 Citation (Scopus)

Abstract

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x∈H, a spectral decomposition is introduced, and for any function f:R→R, we define a corresponding vector-valued function fH(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x∈H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.

Original languageEnglish
Pages (from-to)5766-5783
Number of pages18
JournalNonlinear Analysis, Theory, Methods and Applications
Volume74
Issue number16
DOIs
Publication statusPublished - 2011 Nov 1

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Second-order Cone
Nonsmooth Function
Spectral Decomposition
Vector-valued Functions
Cones
Semismoothness
Hilbert space
Hilbert spaces
Lipschitz Continuity
Complementarity Problem
Differentiability
Decomposition
Smoothness

Keywords

  • Hilbert space
  • Infinite-dimensional second-order cone
  • Strong semismoothness

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x∈H, a spectral decomposition is introduced, and for any function f:R→R, we define a corresponding vector-valued function fH(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x∈H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.",
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AB - Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x∈H, a spectral decomposition is introduced, and for any function f:R→R, we define a corresponding vector-valued function fH(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x∈H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.

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