{
  "abstract": "This paper develops a hyperbolic-majorization preconditioned three-term nonlinear conjugate-gradient framework for nonconvex finite minimax optimization. An analytic symmetric positive definite metric is derived from a global quadratic majorization of the hyperbolic smoothing model and is used simultaneously as a curvature absorber, a preconditioner, and the line-search energy metric. With the displacement \\(s_{k-1}\\) fixed and a variable curvature response \\(b_k\\), an enhanced three-term direction is introduced together with an adaptive parameter \\(μ_k^\\star\\) that is maximal under the requirement that the baseline worst-case metric-energy constant be preserved. The resulting framework yields a Dai--Liao-type conjugacy relation, enhanced sufficient descent, a smoothing-parameter-uniform Armijo lower bound, fixed-smoothing global first-order convergence and complexity, and Clarke-stationary accumulation points under continuation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13754",
  "author_total": 1,
  "authors": [
    {
      "name": "Wenzhe Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6a5zw72ye6paj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13754",
  "fulltext_url": null,
  "id": "6a5zw72ye6paj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13754",
  "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": "W7213270058",
  "pdf_url": "https://arxiv.org/pdf/2609.13754",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6a5zw72ye6paj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13754",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13754"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Preconditioned Three-Term Conjugate Gradient Framework for Nonconvex Finite Minimax Problems",
  "title_text": "Preconditioned Three-Term Conjugate Gradient Framework for Nonconvex Finite Minimax Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/6a5zw72ye6paj",
  "venue": "arXiv",
  "versions": []
}