Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization

Junfeng Yang Nanjing University Xiaoming Yuan Hong Kong Baptist University

Optimization and Control mathscidoc:1701.27010

Mathematics of Computation, 82, (281), 301–329, 2013.1
The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to have closed-form solutions. In this paper, we are interested in the application of the ALM and the ADM for some nuclear norm involved minimization problems. When the resulting subproblems do not have closed-form solutions, we propose to linearize these subproblems such that closed-form solutions of these linearized subproblems can be easily derived. Global convergence results of these linearized ALM and ADM are established under standard assumptions. Finally, we verify the effectiveness and efficiency of these new methods by some numerical experiments.
Convex programming, nuclear norm, low-rank, augmented Lagrangian method, alternating direction method, linearized
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@inproceedings{junfeng2013linearized,
  title={Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization},
  author={Junfeng Yang, and Xiaoming Yuan},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170126171855045715126},
  booktitle={Mathematics of Computation},
  volume={82},
  number={281},
  pages={301–329},
  year={2013},
}
Junfeng Yang, and Xiaoming Yuan. Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization. 2013. Vol. 82. In Mathematics of Computation. pp.301–329. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170126171855045715126.
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