{
  "abstract": "We study the maximum-volume inscribed ellipsoid (MaxIE) problem for a polytope through a geometric iteration based on polarity. Given an interior point, the method forms the shifted polar polytope, computes its minimum-volume covering ellipsoid (MinCE), and then polarizes this covering ellipsoid back to obtain a new inscribed ellipsoid. This procedure was suggested by Khachiyan and Todd in 1993. Prior work studied basic polarity identities and gave an asymptotic-convergence argument for the exact process, but did not furnish a volume-ratio rate. Independently of that asymptotic argument, we develop a new analysis based on the log-volume of the polar MinCE as a potential function. We prove its convexity with an explicit gradient formula and establish global linear contraction of the potential gap along each trajectory. The contraction factor is existential and instance-dependent. This gives both a separate convergence proof and a finite volume-ratio iteration bound. We further analyze an inexact polarity process in which each MinCE subproblem is solved only approximately, and derive sufficient oracle tolerances for producing a prescribed volume approximation. Combining this outer analysis with existing algorithms for MinCE gives conditional arithmetic estimates based on the path-following Newton method, the barycentric coordinate descent, and the away-step Frank-Wolfe method. Numerical experiments compare these oracle choices with two MaxIE baselines and illustrate that performance depends on matching the MinCE solver to the instance geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10888",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaizhao Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/71eaek5gndd21.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10888",
  "fulltext_url": null,
  "id": "71eaek5gndd21",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10888",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7212309466",
  "pdf_url": "https://arxiv.org/pdf/2609.10888",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71eaek5gndd21.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10888",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10888"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Polarity Process for the Maximum-Volume Inscribed Ellipsoid Problem",
  "title_text": "The Polarity Process for the Maximum-Volume Inscribed Ellipsoid Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/71eaek5gndd21",
  "venue": "arXiv",
  "versions": []
}