On the Rate of Convergence of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914414078197760 |
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| author | Foglia, Katherine Rossella Colao, Vittorio |
| author_facet | Foglia, Katherine Rossella Colao, Vittorio |
| contents | The Krasnosel'ski\uı--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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Rate of Convergence of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization Foglia, Katherine Rossella Colao, Vittorio Optimization and Control Functional Analysis The Krasnosel'ski\uı--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. |
| title | On the Rate of Convergence of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization |
| topic | Optimization and Control Functional Analysis |
| url | https://arxiv.org/abs/2510.25363 |