{
  "abstract": "We consider the convex problem $(P): {\\min}_{y\\in \\mathbb{R}^n}\\, f(Ay)+g(y)$, where $A:\\mathbb{R}^n\\to \\mathbb{R}^m$ is a linear operator, $f$ is convex and $(M,ν)$-Hölder smooth on $\\mathbb{R}^m$, and $g$ is proper, closed convex on $\\mathbb{R}^n$, and admits a (generalized) linear minimization oracle (LMO). An active-hull (AH) method calls the LMO in each iteration and outputs a point that lies in the convex hull of all the past LMO returns. We show that under the Euclidean geometry (and in the high-dimensional regime), the optimal LMO complexity for the class of AH methods to solve $(P)$ is $O\\big(M^{{2}/({1+ν})}D^2\\varepsilon^{-{2}/({1+ν})}\\big)$, where $D$ denotes the diameter of the feasible region and $\\varepsilon$ denotes the objective sub-optimality gap. In particular, the optimal LMO complexity is achieved by a single-loop first-order primal-dual splitting method. Since this method requires knowledge of the problem parameters (e.g., $M$, $D$ and $ν$), we propose two meta parameter-search algorithms that aim to resolve this issue. These algorithms are parameter-free, but come with the price of an additional log-factor in their LMO complexities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34101",
  "author_total": 1,
  "authors": [
    {
      "name": "Renbo Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f8xr728m6y7c9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34101",
  "fulltext_url": null,
  "id": "f8xr728m6y7c9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34101",
  "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": "W7214748418",
  "pdf_url": "https://arxiv.org/pdf/2609.34101",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f8xr728m6y7c9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2609.34101",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34101"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On the Optimal Linear Minimization Oracle Complexity of Active-Hull Methods for Minimizing Convex Hölder Smooth Functions",
  "title_text": "On the Optimal Linear Minimization Oracle Complexity of Active-Hull Methods for Minimizing Convex Hölder Smooth Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/f8xr728m6y7c9",
  "venue": "arXiv",
  "versions": []
}