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

Ching-Yu Yang Yu-Lin Chang Jein-Shan Chen

Optimization and Control mathscidoc:1910.43919

Nonlinear Analysis: Theory, Methods & Applications, 74, (16), 5766-5783, 2011.11
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 f H (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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@inproceedings{ching-yu2011analysis,
  title={Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones},
  author={Ching-Yu Yang, Yu-Lin Chang, and Jein-Shan Chen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020224809039534448},
  booktitle={Nonlinear Analysis: Theory, Methods & Applications},
  volume={74},
  number={16},
  pages={5766-5783},
  year={2011},
}
Ching-Yu Yang, Yu-Lin Chang, and Jein-Shan Chen. Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones. 2011. Vol. 74. In Nonlinear Analysis: Theory, Methods & Applications. pp.5766-5783. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020224809039534448.
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