A Semismooth Newton based Augmented Lagrangian Method for Nonsmooth Optimization on Matrix Manifolds

Yuhao Zhou Department of Computer Science and Technology, Tsinghua University, China Chenglong Bao Yau Mathematical Sciences Center, Tsinghua University, China and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, China Chao Ding Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, China Jun Zhu Department of Computer Science and Technology, Tsinghua University, China

Optimization and Control mathscidoc:2206.27001

2021.11
This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and non-negative constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method.
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@inproceedings{yuhao2021a,
  title={A Semismooth Newton based Augmented Lagrangian Method for Nonsmooth Optimization on Matrix Manifolds},
  author={Yuhao Zhou, Chenglong Bao, Chao Ding, and Jun Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613165639466827349},
  year={2021},
}
Yuhao Zhou, Chenglong Bao, Chao Ding, and Jun Zhu. A Semismooth Newton based Augmented Lagrangian Method for Nonsmooth Optimization on Matrix Manifolds. 2021. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613165639466827349.
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