A Bregman projection method for approximating fixed points of quasi-Bregman nonexpansive mappings

Abstract

We introduce an abstract algorithm that aims to find the Bregman projection onto a closed convex set. As an application, the asymptotic behavior of an iterative method for finding a fixed point of a quasi-Bregman nonexpansive mapping with the fixed-point closedness property is analyzed. We also show that our result is applicable to Bregman subgradient projectors.

Keywords:

• Bregman projection
• Bregman subgradient projector
• fixed point
• Legendre function
• Moreau envelope
• quasi-Bregman nonexpansive

AMSSubject Classifications:

• Primary: 47H09
• Secondary: 52B55
• 65K10
• 90C25

Acknowledgments

We thank the referee for some helpful suggestions and pertinent comments. HHB was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Canada Research Chair Program. JC was partially supported by the Natural Science Foundation of China, the Doctor Fund of Southwest University, and the Fundamental Research Fund for the Central Universities. XW was partially supported by the Natural Sciences and Engineering Research Council of Canada.

Notes

Dedicated to Boris Mordukhovich on the occasion of his 65th birthday.