The Bregman Ball-Picard Method: A Bregman-Distance Extension of the Ball-Picard Method for Monotone Inclusions
Fixed Point Methods and Optimization, Volume 3, Issue 3, December 2026, Pages 174–180
Amnay El Amri
Hassan II University, Facult\'e des Sciences Ben M'Sik, Avenue Cdt Driss El Harti, BP 7955, 20670 Casablanca, Morocco
Abdellatif Moudafi
Aix-Marseille Université, Laboratoire d'Informatique et Systèmes (LIS), UMR CNRS 7020, Faculté des Sciences - Campus de Saint-Jérôme, 52 Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France
Abstract
The recently introduced Ball-Picard Method extends the Ball-Proximal Point Method from convex minimization to the general fixed-point problem for a nonexpansive operator, by replacing the unconstrained proximal step with a step constrained to a Euclidean ball around the current iterate. We extend this method to the Bregman-distance setting, replacing the Euclidean ball and norm with a Bregman ball and a Bregman distance. For nonexpansive operators given by Bregman resolvents of maximal monotone operators – covering in particular Bregman-proximal minimization – we prove that the resulting implicit iteration is well posed and enjoys a Fejér-type decrease property, together with the corresponding finite- and weak-convergence guarantees, and we verify that the Euclidean case is recovered as a special instance. We also identify precisely where a direct extension to arbitrary Bregman-nonexpansive operators breaks down, leaving it as an open problem, and we briefly discuss how the implicit iteration can be solved in practice. The results specialize correctly to Bregman-proximal minimization and recover the original Euclidean theorem exactly when the Bregman distance is taken to be the squared Euclidean norm, confirming that the generalization is genuine and non-degenerate.
Cite this Article as
Amnay El Amri and Abdellatif Moudafi, The Bregman Ball-Picard Method: A Bregman-Distance Extension of the Ball-Picard Method for Monotone Inclusions, Fixed Point Methods and Optimization, 3(3), 174–180, 2026