{
  "abstract": "We consider convex optimization with nonlinear inequality constraints and develop a primal-dual multiplier framework that is consistent in continuous and discrete time. We first propose continuous-time dynamics with Nesterov-type vanishing damping $α/t$, together with compatible extrapolations of the dual variable and the nonlinear constraint mapping. Under convexity assumptions and $α\\geq3$, we establish $\\mathcal O(t^{-2})$ convergence rates for both nonlinear feasibility and the objective residual. In the noncritical regime $α>3$, with an admissible choice of the extrapolation parameter, we further prove that the entire primal-dual trajectory converges to a KKT pair and sharpen both continuous-time estimates to $o(t^{-2})$. We then derive an inexact accelerated primal-dual algorithm through a compatible discretization of a perturbed version of the dynamic. For composite convex objectives, a weighted summability condition on the primal inexactness yields $\\mathcal O(k^{-2})$ rates for feasibility and the objective residual. In the corresponding noncritical regime, the discrete primal-dual sequence converges to a KKT pair and both residual estimates improve to $o(k^{-2})$. Thus the continuous and discrete results exhibit matching accelerated rates and matching asymptotic improvements. To the best of our knowledge, this is the first Nesterov-type primal-dual multiplier framework for convex optimization with nonlinear inequality constraints.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01415",
  "author_total": 1,
  "authors": [
    {
      "name": "Xin He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24f0szfz2h5za.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01415",
  "fulltext_url": null,
  "id": "24f0szfz2h5za",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01415",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7206180691",
  "pdf_url": "https://arxiv.org/pdf/2609.01415",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24f0szfz2h5za.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.01415",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01415"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Accelerated primal-dual dynamics and algorithms for convex optimization with nonlinear inequality constraints",
  "title_text": "Accelerated primal-dual dynamics and algorithms for convex optimization with nonlinear inequality constraints",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/24f0szfz2h5za",
  "venue": "arXiv",
  "versions": []
}