A convergence rate of the proximal point algorithm in Banach spaces

Abstract

We consider the convergence rate of the proximal point algorithm (PPA) for finding a minimizer of proper lower semicontinuous convex functions. In the Hilbert space setting, Güler showed that the big-O rate of the PPA can be improved to little-o when the sequence generated by the algorithm converges strongly to a minimizer. In this paper, we establish little-o rate of the PPA in Banach spaces without requiring this assumption. Then we apply the result to give new results on the convergence rate for sequences of alternating and averaged projections.

Keywords:

• Proximal point algorithm
• convergence rate
• convex minimization problem
• Banach space
• alternating projections
• averaged projections

Acknowledgements

The author thanks the editors and the anonymous referees for their comments and suggestions which improved the presentation of this paper. The author is grateful to Professors W. Takahashi of Tokyo Institute of Technology, D. Kuroiwa of Shimane university and Li Xu of Akita Prefectural University for their helpful support.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported in part by the Ministry of Education, Culture, Sports, Science, and Technology [grant number 16K05280].