On the proximal point method in Hadamard spaces

Abstract

Proximal point method is one of the most influential procedure in solving nonlinear variational problems. It has recently been introduced in Hadamard spaces for solving convex optimization, and later for variational inequalities. In this paper, we study the general proximal point method for finding a zero point of a maximal monotone set-valued vector field defined on a Hadamard space and valued in its dual. We also give the relation between the maximality and Minty’s surjectivity condition, which is essential for the proximal point method to be well-defined. By exploring the properties of monotonicity and the surjectivity condition, we were able to show under mild assumptions that the proximal point method converges weakly to a zero point. Additionally, by taking into account the metric subregularity, we obtained the local strong convergence in linear and super-linear rates.

Keywords:

• Proximal point method
• zero point problem
• optimization
• variational inequality
• Hadamard space
• maximal monotonicity
• Minty’s surjectivity condition
• quasilinearization

Acknowledgements

The authors are grateful to referee’s and editor’s valuable comments and suggestions, which significantly improved the results in this paper. Parin Chaipunya would like to thank the Thailand Research Fund (TRF) and King Mongkut’s University of Technology Thonburi (KMUTT) for their joint support through the Royal Golden Jubilee PhD Program [grant number PHD/0045/2555]. Poom Kumam would like to thank the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) under the TRF Research Scholar Award (Grant No. RSA6080047) for financial support.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project was partially supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT. The first author was supported jointly by the Thailand Research Fund (TRF) and King Mongkut’s University of Technology Thonburi (KMUTT) through the Royal Golden Jubilee PhD Program [grant number PHD/0045/2555]. Moreover, Poom Kumam would like to thanks the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) under the TRF Research Scholar Award [grant number RSA6080047] for financial support.