From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems

[2203.04869] From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Optimization and Control arXiv:2203.04869 (math) [Submitted on 9 Mar 2022 ( v1 ), last revised 15 Apr 2026 (this version, v3)] Title: From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems Authors: Quoc Tran-Dinh View a PDF of the paper titled From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems, by Quoc Tran-Dinh View PDF HTML (experimental) Abstract: We derive an equivalent form of Halpern's fixed-point iteration scheme for solving a co-coercive equation (also called a root-finding problem), which can be viewed as a Nesterov's accelerated interpretation. We show that one method is equivalent to another via a simple transformation, leading to a straightforward convergence proof for Nesterov's accelerated scheme. Alternatively, we directly establish convergence rates of Nesterov's accelerated variant, and as a consequence, we obtain a new convergence rate of Halpern's fixed-point iteration. Next, we apply our results to different methods to solve monotone inclusions, where our convergence guarantees are applied. Since the gradient/forward scheme requires the co-coerciveness of the underlying operator, we derive new Nesterov's accelerated variants for both recent extra-anchored gradient and past-extra anchored gradient methods in the literature. These variants alleviate the co-coerciveness condition by only assuming the monotonicity and Lipschitz continuity of the underlying operator. Interestingly, our new Nesterov's accelerated interpretation of the past-extra anchored gradient method involves two past-iterate correction terms. This formulation is expected to guide us developing new Nesterov's accelerated methods for minimax problems and their continuous views without co-coericiveness. We test our theoretical results on two numerical examples, where the actual convergence rates match well the theoretical ones up to a constant factor. Comments: 32 pages, 2 figures Subjects: Optimization and Control (math.OC) Report number: UNC-STOR-03042022 Cite as: arXiv:2203.04869 [math.OC] (or arXiv:2203.04869v3 [math.OC] for this version) https://doi.org/10.48550/arXiv.2203.04869 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Quoc Tran-Dinh [ view email ] [v1] Wed, 9 Mar 2022 16:45:10 UTC (107 KB) [v2] Tue, 7 Oct 2025 14:29:06 UTC (120 KB) [v3] Wed, 15 Apr 2026 13:06:22 UTC (121 KB) Full-text links: Access Paper: View a PDF of the paper titled From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems, by Quoc Tran-Dinh View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2022-03 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

Related documents

• Memory-Time Trade-offs for Floating-Point Compression in Industrial Hierarchical Matrix Solvers#291691
• Beyond Compression: Training Latent Representations for Stable Long-Horizon Rollout in Neural Surrogate Solvers#1078269
• A Lower Bound for the Heavy-Ball Method on Smooth Convex Functions#985989
• Small Models, Strong Priors: Architectural Inductive Bias for Parameter-Efficient Neural PDE Solvers#224524
• Accurate and Scalable Matrix Mechanisms via Divide and Conquer#2782