{"ID":2848513,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.25363","arxiv_id":"2510.25363","title":"On the Rate of Convergence of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization","abstract":"The Krasnosel'skiĭ--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 $CAT(κ)$ 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.","short_abstract":"The Krasnosel'skiĭ--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 $CAT(κ)$ and, more generally, geodesic spaces. Convergence results and convergence rate estimates in these nonli...","url_abs":"https://arxiv.org/abs/2510.25363","url_pdf":"https://arxiv.org/pdf/2510.25363v2","authors":"[\"Katherine Rossella Foglia\",\"Vittorio Colao\"]","published":"2025-10-29T10:36:25Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.FA\"]","methods":"[]","has_code":false}