{
  "abstract": "In this paper, we study a Tikhonov-regularized second-order primal--dual dynamical system for smooth nonlinear convex--concave saddle-point problems in real Hilbert spaces. The dynamical system incorporates a vanishing damping coefficient $α/t$, a Hessian-driven term, and a critical Tikhonov regularization coefficient $c/t^2$. By constructing suitable Lyapunov functions, we establish strong convergence to the minimum-norm saddle point and fast convergence rates under the same parameter conditions, without imposing any eventual inside/outside-ball condition. Specifically, the primal--dual trajectory converges strongly to the minimum-norm saddle point, while the primal--dual gap and the velocity norm decay at rates $o(t^{-2})$ and $o(t^{-1})$, respectively. Furthermore, we propose an inexact primal--dual algorithm with a computable relative residual criterion and prove strong convergence of the generated sequence to the same minimum-norm saddle point, together with an $o(k^{-2})$ convergence rate for the primal--dual gap. Finally, numerical experiments on nonlinear and bilinear problems of different dimensions illustrate the minimum-norm selection property and compare the performance of the proposed dynamical system and algorithm with related methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39133",
  "author_total": 2,
  "authors": [
    {
      "name": "Hong-lu Li"
    },
    {
      "name": "Yi-bin Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9z3v4mabbehse.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39133",
  "fulltext_url": null,
  "id": "9z3v4mabbehse",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39133",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39133",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9z3v4mabbehse.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39133",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39133"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Tikhonov-Regularized Dynamics and Inexact Algorithms for Nonlinear Saddle-Point Problems",
  "title_text": "Tikhonov-Regularized Dynamics and Inexact Algorithms for Nonlinear Saddle-Point Problems",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/9z3v4mabbehse",
  "venue": "arXiv",
  "versions": []
}