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A proximal gradient descent method for the extended second-order cone linear complementarity problem

Shaohua Pan Jein-Shan Chen

Optimization and Control mathscidoc:1910.43913

Journal of Mathematical Analysis and Applications, 366, (1), 164-180, 2010.6
We consider an extended second-order cone linear complementarity problem (SOCLCP), including the generalized SOCLCP, the horizontal SOCLCP, the vertical SOCLCP, and the mixed SOCLCP as special cases. In this paper, we present some simple second-order cone constrained and unconstrained reformulation problems, and under mild conditions prove the equivalence between the stationary points of these optimization problems and the solutions of the extended SOCLCP. Particularly, we develop a proximal gradient descent method for solving the second-order cone constrained problems. This method is very simple and at each iteration makes only one Euclidean projection onto second-order cones. We establish global convergence and, under a local Lipschitzian error bound assumption, linear rate of convergence. Numerical comparisons are made with the limited-memory BFGS method for the
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@inproceedings{shaohua2010a,
  title={A proximal gradient descent method for the extended second-order cone linear complementarity problem},
  author={Shaohua Pan, and Jein-Shan Chen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020224619676645442},
  booktitle={Journal of Mathematical Analysis and Applications},
  volume={366},
  number={1},
  pages={164-180},
  year={2010},
}
Shaohua Pan, and Jein-Shan Chen. A proximal gradient descent method for the extended second-order cone linear complementarity problem. 2010. Vol. 366. In Journal of Mathematical Analysis and Applications. pp.164-180. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020224619676645442.
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