{
  "abstract": "Strongly convex minimization and its natural indefinite extension to strongly convex--strongly concave minimax problems combine quantitative control of curvature with a prescribed curvature orientation. We disentangle these two roles by retaining uniform nondegeneracy alone: curvature remains uniformly separated from zero but may have either sign, with no prescribed positive--negative splitting. Surprisingly, a large part of the familiar theory nevertheless re-emerges. We first derive an intrinsic formulation through first-order secant inequalities, making the gradient on $\\mathbb{R}^d$ a global bi-Lipschitz homeomorphism and yielding a unique stationary point. We then pair signed Moreau envelopes to construct a smooth scalar merit that recovers the missing descent geometry at both zeroth and first order, and the resulting paired proximal descent method achieves global linear convergence with dimension-free first-order oracle complexity. Meanwhile, we show that this tractability can break down on restricted domains: merely assuming the existence of a stationary point in the domain may lead to the curse of dimensionality, even with access to an infinite-order oracle. To overcome this information barrier, we introduce certified feasibility, an observable localization condition that enables feasible continuation. Together, these results establish a first-order optimization theory under uniform nondegeneracy that spans geometry, computation, and information.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06946",
  "author_total": 3,
  "authors": [
    {
      "name": "Hua Su"
    },
    {
      "name": "Lei Zhang"
    },
    {
      "name": "Jin Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c0y9swp0a2yp2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06946",
  "fulltext_url": null,
  "id": "c0y9swp0a2yp2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06946",
  "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",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7212052874",
  "pdf_url": "https://arxiv.org/pdf/2609.06946",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c0y9swp0a2yp2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06946",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06946"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "First-Order Optimization under Uniform Nondegeneracy: Geometry, Computation, and Information",
  "title_text": "First-Order Optimization under Uniform Nondegeneracy: Geometry, Computation, and Information",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/c0y9swp0a2yp2",
  "venue": "arXiv",
  "versions": []
}