{
  "abstract": "Optimal first-order methods on non-Euclidean domains such as the $\\ell_1$ ball $B_1^n(R)=\\{x\\in\\mathbb R^n:\\|x\\|_1\\le R\\}$ pair the prox-function with the norm in which smoothness is measured. When the gradient is $L$-Lipschitz in the Euclidean norm only, the accelerated method with a Euclidean prox-setup reduces the functional gap $f(x_N)-\\min_{B_1^n(R)}f$ to $O(LR^2/N^2)$ after $N$ first-order queries (an entropic $\\ell_1$ prox-setup replaces $L$ by the $\\ell_1\\to\\ell_\\infty$ constant $L_1\\le L$, at the cost of a factor $\\log n$ and with the same exponent). The lower bound of Guzmán and Nemirovski is of order $LR^2/N^3$, and whether the upper bound can be brought down to that order is a question of A. S. Nemirovski. We show that for this mismatched problem the minimax value of the gap is of order $LR^2/N^3$ up to logarithmic factors, for deterministic and for randomized methods alike, and already in dimension proportional to the number of queries. The upper bound is attained by a Steiner-point level method: every supporting hyperplane seen so far is kept as a level cut, and the next query is made near the Steiner point of the resulting localization polytope. The analysis rests on a single geometric fact: along nested subsets of the ball the Steiner points travel a distance that is polylogarithmic in the dimension, in contrast to $\\sqrt n$ for the Euclidean ball. All queries stay feasible, and a randomized selector keeps the internal work polynomial. The same geometry yields optimal rates for nonsmooth objectives, Hölder gradients, higher-order oracles and Lipschitz monotone operators, the last with a matching deterministic lower bound. For convex quadratics, a curvature-learning method attains the optimal rate on every $\\ell_p$ ball with $1\\le p<2$ and without logarithmic loss. Experiments confirm the predicted $N^{-3}$ behaviour on objectives that are hard for Euclidean methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32910",
  "author_total": 7,
  "authors": [
    {
      "name": "Timofei Loginov"
    },
    {
      "name": "Alexander Gasnikov"
    },
    {
      "name": "Yuriy Dorn"
    },
    {
      "name": "Aleksandr Shestakov"
    },
    {
      "name": "Nazarii Tupitsa"
    },
    {
      "name": "Osman Osmanov"
    },
    {
      "name": "Darina Dvinskikh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86z69thqwf768.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32910",
  "fulltext_url": null,
  "id": "86z69thqwf768",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32910",
  "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": "W7214709109",
  "pdf_url": "https://arxiv.org/pdf/2609.32910",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86z69thqwf768.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32910",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32910"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Complexity of Convex Optimization with Mismatched Geometry",
  "title_text": "The Complexity of Convex Optimization with Mismatched Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/86z69thqwf768",
  "venue": "arXiv",
  "versions": []
}