Tikhonov-Regularized Second-Order Primal-Dual Dynamics for Convex-Concave Bilinear Saddle Point Problems

[2601.23120] Tikhonov-Regularized Second-Order Primal-Dual Dynamics for Convex-Concave Bilinear Saddle Point Problems Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2601.23120 (math) [Submitted on 30 Jan 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Tikhonov-Regularized Second-Order Primal-Dual Dynamics for Convex-Concave Bilinear Saddle Point Problems Authors: Bohan Zhang , Xiaojun Zhang View PDF HTML (experimental) Abstract: This paper develops a Lyapunov-based coefficient-compatible framework for a class of second-order primal-dual dynamical systems with Tikhonov regularization. Unlike existing accelerated inertial primal-dual dynamics, whose analyses are typically carried out under prescribed damping profiles, the proposed framework characterizes the interplay among the viscous damping coefficient, the time-scaling coefficient, the linear extrapolation parameter, and the Tikhonov regularization parameter through differential compatibility conditions, thereby accommodating time-varying coefficients beyond prescribed damping profiles. Within this framework, we investigate two cases in which the Tikhonov regularization parameter decays to zero at different rates. When the Tikhonov regularization parameter decays to zero relatively rapidly, the fast convergence rate of the primal-dual gap is preserved, whereas a slower decay rate yields refined asymptotic convergence estimates. We further show that a sufficiently slow decay ensures strong convergence of the trajectories to the minimum-norm saddle point. To establish the latter property, we introduce a moving-saddle Lyapunov framework centered at the instantaneous regularized saddle point and derive verifiable coefficient conditions ensuring the required tracking behavior. Finally, three numerical examples are presented to illustrate the theoretical results. Subjects: Optimization and Control (math.OC) ; Dynamical Systems (math.DS) Cite as: arXiv:2601.23120 [math.OC] (or arXiv:2601.23120v2 [math.OC] for this version) https://doi.org/10.48550/arXiv.2601.23120 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Bohan Zhang [ view email ] [v1] Fri, 30 Jan 2026 16:08:19 UTC (793 KB) [v2] Thu, 27 Aug 2026 09:39:30 UTC (368 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2026-01 Change to browse by: math math.DS 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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