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.
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.