New results on the proximal point algorithm in non-positive curvature metric spaces

Abstract

The subject of this paper is the inexact proximal point algorithm of usual and Halpern type in non-positive curvature metric spaces. We study the convergence of the sequences given by the inexact proximal point algorithm with non-summable errors. We also prove the strong convergence of the Halpern proximal point algorithm to a minimum point of the convex function. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces.

Keywords:

• Convex minimization
• proximal point algorithm
• Halpern method
• Hadamard space
• resolvent

Acknowledgements

The authors are grateful to the referees for their careful reading and valuable comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was in part supported by a grant from University of Zanjan [No. 9244].