Abstract strongly convergent variants of the proximal point algorithm

TL;DR

This paper establishes strong convergence results for Halpern-type and Tikhonov-type proximal point algorithms in CAT(0) spaces and provides explicit rates of metastability using proof mining.

Contribution

It introduces an abstract convergence framework for these algorithms in CAT(0) spaces and derives uniform, computable metastability rates.

Findings

Proved strong convergence of the algorithms in CAT(0) spaces.

Derived explicit metastability rates for the iterative processes.

Applied proof mining techniques to obtain quantitative convergence information.

Abstract

We prove an abstract form of the strong convergence of the Halpern-type and Tikhonov-type proximal point algorithms in CAT(0) spaces. In addition, we derive uniform and computable rates of metastability (in the sense of Tao) for these iterations using proof mining techniques.