{
  "abstract": "We study how Difference of Convex (DC) decomposition affects cutting-plane approximation in global optimization. For a smooth decomposition $f = g-h$, let $d$ be the dimension of the span of all Hessian ranges of $g$ over the convex hull of a compact feasible set. The smallest attainable dimension equals the minimum rank of a constant positive semidefinite matrix majorizing the Hessians of $f$ on that convex hull, and is attained by a quadratic first component. Retaining earlier gradient cuts and accepting new cuts only when the model error exceeds $ε$ gives an $O(ε^{-d/2})$ bound on accepted cuts. For an adaptive global DC method, this also bounds the number of subproblems solved exactly and globally. Under nondegeneracy, the exponent is sharp for uniform cutting-plane approximation of quadratic first components. A structured example reduces the certified exponent from $n/2$ to $1/2$ and the global subproblem to one variable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28319",
  "author_total": 1,
  "authors": [
    {
      "name": "Fahaar M. Pirani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fn5j0kjd3mhek.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28319",
  "fulltext_url": null,
  "id": "fn5j0kjd3mhek",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28319",
  "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": "W7214171654",
  "pdf_url": "https://arxiv.org/pdf/2609.28319",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fn5j0kjd3mhek.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.28319",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28319"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "DC decompositions and cutting-plane approximation: geometry and complexity",
  "title_text": "DC decompositions and cutting-plane approximation: geometry and complexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/fn5j0kjd3mhek",
  "venue": "arXiv",
  "versions": []
}