{
  "abstract": "We study convex optimization over compact convex sets for twice differentiable objective functions with positive definite Hessians, assuming access to the feasible set through a linear minimization oracle. Existing second-order Frank--Wolfe methods either provide only a local linear rate over general convex sets or achieve global linear and local quadratic convergence only over polytopes. We propose damped Newton Frank--Wolfe methods that approximately solve constrained damped Newton subproblems using Frank--Wolfe or away-step Frank--Wolfe inner iterations. We develop residual backtracking, using a root Newton stepsize as a safe reference, together with a switching rule that eventually takes full steps. Under Hölder smoothness assumptions for $p$th derivatives, we establish global linear convergence and local convergence with Q-order at least $1+ν$ ($p=2$) and local quadratic convergence ($p=3$). Experiments on matrix sensing and ridge-regularized logistic regression show that the residual-backtracking variant is consistently robust and often achieves the best performance among the tested first- and second-order baselines.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34290",
  "author_total": 2,
  "authors": [
    {
      "name": "Shota Takahashi"
    },
    {
      "name": "Akiko Takeda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bzdz1agx2rxj2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34290",
  "fulltext_url": null,
  "id": "bzdz1agx2rxj2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34290",
  "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",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214768147",
  "pdf_url": "https://arxiv.org/pdf/2609.34290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bzdz1agx2rxj2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Beyond Polytopes: Damped Newton Frank--Wolfe Methods over Compact Convex Sets",
  "title_text": "Beyond Polytopes: Damped Newton Frank--Wolfe Methods over Compact Convex Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bzdz1agx2rxj2",
  "venue": "arXiv",
  "versions": []
}