{
  "abstract": "We extend a $d$th-order Newton method for unconstrained optimization by Ahmadi, Chaudhry, and Zhang [Advances in Mathematics, 452:109808] to optimization with SOS-convex polynomial constraints. Consider the problem of minimizing a smooth function $f:\\mathbb{R}^{n}\\to\\mathbb{R}$ subject to SOS-convex polynomial constraints. Given an iterate $x\\in\\mathbb{R}^n$, Ahmadi et al. define the next iterate $x^{+}$ as the minimizer of the $d$th-order Taylor expansion of $f$ at $x$ with a regularization term of degree $d^{\\prime}$, where $d^{\\prime}$ is the smallest even number greater than $d$, chosen such that this polynomial is SOS-convex, subject to the constraints. Constructing this polynomial and minimizing it subject to the constraints can both be reduced in time polynomial in $n$ to a semidefinite program (SDP). We prove that, if $f$ is strongly convex and the tensor of the $d$th-order partial derivatives of $f$ is Lipschitz continuous, then our method converges locally to the optimal solution $x^{\\ast}$ with order $d$. We further prove that, under certain constraint qualifications, the set of active constraints at $x^{\\ast}$ is identified locally in a single iteration. Next, we study the worst-case performance of the third-order Newton method in the unconstrained setting for two classes of univariate $f$ using performance estimation. Finally, we extend a globally convergent modification of the $d$th-order Newton method to the setting of SOS-convex polynomial constraints.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01001",
  "author_total": 2,
  "authors": [
    {
      "name": "Lucas ter Voert"
    },
    {
      "name": "Etienne de Klerk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vantdbzpq2s9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01001",
  "fulltext_url": null,
  "id": "5vantdbzpq2s9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01001",
  "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",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7206172116",
  "pdf_url": "https://arxiv.org/pdf/2609.01001",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vantdbzpq2s9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01001",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01001"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Further analysis and extension of the higher-order Newton method of Ahmadi, Chaudhry, and Zhang",
  "title_text": "Further analysis and extension of the higher-order Newton method of Ahmadi, Chaudhry, and Zhang",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/5vantdbzpq2s9",
  "venue": "arXiv",
  "versions": []
}