General parameterized proximal point algorithm with applications in statistical learning

ABSTRACT

In the literature, there are a few researches to design some parameters in the proximal point algorithm (PPA), especially for the multi-objective convex optimizations. Introducing some parameters to PPA can make it more flexible and attractive. Mainly motivated by our recent work [Bai et al. A parameterized proximal point algorithm for separable convex optimization. Optim Lett. (2017) doi:10.1007/s11590-017-1195-9], in this paper we develop a general parameterized PPA with a relaxation step for solving the multi-block separable structured convex programming. By making use of the variational inequality and some mathematical identities, the global convergence and the worst-case $\mathcal{O}\left(1/t\right)$ convergence rate of the proposed algorithm are established. Preliminary numerical experiments on solving a sparse matrix minimization problem from statistical learning validate that our algorithm is more efficient than several state-of-the-art algorithms.

Keywords:

• Structured convex programming
• proximal point algorithm
• relaxation step
• complexity
• statistical learning

2010 AMS Subject Classifications:

• 65C60
• 65Y20
• 90C25

Acknowledgements

The authors wish to thank the Editor-in-Chief Prof. Choi-Hong Lai and the anonymous referees for providing their valuable suggestions, which have significantly improved the quality of our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. http://web.stanford.edu/∼boyd/papers/admm/covsel/covsel_example.html.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China under grant 11671318 and the Natural Science Foundation of Fujian Province under grant 2016J01028.