{
  "abstract": "In this paper, we study a Tikhonov-regularized mixed-order primal--dual dynamical system with implicit Hessian damping for linearly constrained convex optimization problems in finite-dimensional Euclidean spaces, where the primal equation is second order and incorporates the viscous damping term \\(δ\\sqrt{\\varepsilon(t)}\\,\\dot x(t)\\), whereas the multiplier equation remains first order. By constructing a new class of energy functions, for a general Tikhonov regularization coefficient \\(\\varepsilon(t)\\), we prove the strong convergence of the primal trajectory and derive fast convergence rates under the same parameter assumptions, without imposing any eventual inside/outside-ball condition. More precisely, the primal trajectory converges to the minimum-norm solution, and the multiplier converges to a compatible KKT multiplier, while the convergence rates of the Lagrangian gap, feasibility violation, and objective residual are \\(o(\\varepsilon(t))\\), and the convergence rate of the velocity norm is \\(o(\\sqrt{\\varepsilon(t)})\\). For the critical case \\(\\varepsilon(t)=c/t^2\\), in which the damping coefficient \\(δ\\sqrt{\\varepsilon(t)}\\) reduces to \\(δ\\sqrt{c}/t\\), we establish the sharper convergence rates \\(o(t^{-2})\\) for the Lagrangian gap, feasibility violation, and objective residual, together with \\(o(t^{-1})\\) for the velocity norm, which improve the corresponding \\(O(t^{-2})\\) and \\(O(t^{-1})\\) decay estimates obtained in the related literature. Most importantly, when the proposed dynamical system is specialized to the finite-dimensional unconstrained setting, our analysis answers the open question on strong convergence in this critical regime posed by Attouch and László [Math. Methods Oper. Res., 99 (2024), pp.~307--347].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06420",
  "author_total": 2,
  "authors": [
    {
      "name": "Hong-lu Li"
    },
    {
      "name": "Yi-bin Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpgp4xb97q6s3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06420",
  "fulltext_url": null,
  "id": "fpgp4xb97q6s3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06420",
  "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",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7212079368",
  "pdf_url": "https://arxiv.org/pdf/2609.06420",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpgp4xb97q6s3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06420",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06420"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions",
  "title_text": "Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fpgp4xb97q6s3",
  "venue": "arXiv",
  "versions": []
}