On the Rate of Convergence of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization

TL;DR

This paper extends convergence rate analysis of iterative fixed point methods from Banach spaces to $CAT(0)$ spaces and introduces a Halpern-type optimizer for hyperbolic optimization, broadening the scope of nonlinear fixed point theory.

Contribution

It adapts proof techniques from linear spaces to $CAT(0)$ spaces and proposes a new Halpern-type optimizer for hyperbolic optimization problems.

Findings

Recovered asymptotic regularity bounds in $CAT(0)$ spaces.

Extended convergence analysis techniques to nonlinear settings.

Introduced a Halpern--type optimizer for hyperbolic optimization.

Abstract

The Krasnosel'ski\u{\i}--Mann and Halpern iterations are classical schemes for approximating fixed points of nonexpansive mappings in Banach spaces, and have been widely studied in more general frameworks such as $CAT(\kappa)$ and, more generally, geodesic spaces. Convergence results and convergence rate estimates in these nonlinear settings are already well established. The contribution of this paper is twofold: first, we extend to complete $CAT(0)$ spaces proof techniques originally developed in the linear setting of Banach and Hilbert spaces, thereby recovering the same asymptotic regularity bounds; second, we introduce a Halpern--type optimizer for hyperbolic optimization as a nonlinear counterpart of the Euclidean HalpernSGD scheme.