Proximal Nonlinear Conjugate Gradient Methods for Composite Optimization

[2604.11150] Proximal Nonlinear Conjugate Gradient Methods for Composite Optimization Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2604.11150 (math) [Submitted on 13 Apr 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Proximal Nonlinear Conjugate Gradient Methods for Composite Optimization Authors: Shodai Hamana , Yasushi Narushima View PDF HTML (experimental) Abstract: The nonlinear conjugate gradient methods are known to be an effective approach for standard unconstrained optimization problems especially for large-scale this http URL paper proposes a proximal nonlinear conjugate gradient method, which extends the nonlinear conjugate gradient methods to composite objective functions, namely, the sum of a smooth nonconvex function and a nonsmooth convex function, and its extension to the case where the nonsmooth function is weakly convex. The proposed method uses the \textit{forward-backward residual} which is defined by using the proximal mapping instead of the gradient and determines the search direction based on the three-term Hestenes-Stiefel (HS) formula. We establish the global convergence of the proposed algorithm under standard assumptions for both convex and weakly convex nonsmooth functions, and analyze its convergence rate in terms of this http URL addition, when the smooth term is strongly convex, we prove that the generated sequence converges to the optimal solution and establish its convergence rate. Finally, numerical experiments show that the proposed method is stable and achieves better performance than existing methods in both convex and nonconvex settings. Comments: 36pages, 8figures Subjects: Optimization and Control (math.OC) ; Numerical Analysis (math.NA) MSC classes: 90C30, 90C25, 90C26 Cite as: arXiv:2604.11150 [math.OC] (or arXiv:2604.11150v2 [math.OC] for this version) https://doi.org/10.48550/arXiv.2604.11150 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yasushi Narushima [ view email ] [v1] Mon, 13 Apr 2026 08:11:34 UTC (500 KB) [v2] Thu, 27 Aug 2026 03:30:42 UTC (502 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2026-04 Change to browse by: cs cs.NA math math.NA References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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