{
  "abstract": "We study the deterministic oracle complexity of finding approximate solutions to composite monotone inclusion problems, formed by the sum of a smooth single-valued monotone operator and a maximally monotone set-valued operator, under the tangent-residual criterion. We introduce the Anchored Extra-Proximal (AEP) framework, which combines an anchored extrapolation step with an inexact anchored proximal update satisfying a relative-error condition. The framework recovers the composite Fast Extragradient method in the first-order setting and yields natural second- and higher-order extensions by replacing the operator in the implicit update with its Taylor approximation at the extrapolated point. For every $p\\geq 2$, assuming that the $(p-1)$th derivative of the single-valued operator is Lipschitz continuous, we combine this construction with a bisection line search to obtain a $p$th-order method that finds a point with tangent residual at most $\\varepsilon$ in $\\widetilde{O}(\\varepsilon^{-2/(3p-1)})$ oracle calls. This improves all prior upper bounds for $p$th-order methods: in particular, it improves the previous best-known $\\widetilde{O}(\\varepsilon^{-1/p})$ tangent-residual complexity as well as the classical $O(\\varepsilon^{-2/(p+1)})$ bound of higher-order hybrid proximal extragradient methods under the weaker duality-gap criterion. We complement this result with a worst-case lower bound of $Ω(\\varepsilon^{-2/(3p-1)})$ for every deterministic algorithm in the $p$th-order oracle model, without restricting the algorithm to tensor steps or any other prescribed update structure. Thus, the proposed method attains the optimal dependence on $\\varepsilon$, up to logarithmic factors, for all $p\\geq2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30212",
  "author_total": 2,
  "authors": [
    {
      "name": "Ruichen Jiang"
    },
    {
      "name": "TaeHo Yoon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/847ynz007v4z4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30212",
  "fulltext_url": null,
  "id": "847ynz007v4z4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30212",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7214279250",
  "pdf_url": "https://arxiv.org/pdf/2609.30212",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/847ynz007v4z4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30212",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30212"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Anchored Extra-Proximal Methods: Optimal Higher-Order Methods for Monotone Inclusion Problems",
  "title_text": "Anchored Extra-Proximal Methods: Optimal Higher-Order Methods for Monotone Inclusion Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:21Z",
  "url": "https://research.draconitic.com/paper/847ynz007v4z4",
  "venue": "arXiv",
  "versions": []
}