{
  "abstract": "Minimax optimization is a fundamental framework in machine learning, robust optimization, and game theory, yet finding first-order stationary points of general nonconvex-nonconcave minimax problems remains challenging without additional structural assumptions. Existing guarantees often rely on global PL- or KL-type conditions that connect max-player stationarity to global inner optimality, or on Minty-type conditions that impose a global relation on the game gradient field relative to a reference solution; local KL variants relax the former requirement but typically require initialization and tracking within a near-optimal region. Such conditions may be difficult to satisfy in many applications. In contrast, we develop a first-order method that finds an $ε$-stationary point within $\\widetilde{O}(ε^{-2})$ first-order iterations under a local inverse-Lipschitz regularity condition around approximate max-player stationary points, together with a compactness condition on a penalty sublevel set. Our condition places no optimality requirement on stationary points of the inner maximization problem: they need not be globally, or even locally, maximizing. We further provide sufficient conditions for the required regularity. In the unconstrained setting, it follows from uniform nonsingularity of the maximization-variable Hessian near stationary points; for constrained upper Moreau envelopes, it follows from standard KKT regularity conditions. These results establish first-order complexity guarantees for classes of nonconvex-nonconcave minimax problems not covered by the above PL-, KL-, or Minty-type frameworks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32970",
  "author_total": 3,
  "authors": [
    {
      "name": "Mengxuan Dong"
    },
    {
      "name": "Yao Yao"
    },
    {
      "name": "Jiawei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2k1txcraxrhmk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32970",
  "fulltext_url": null,
  "id": "2k1txcraxrhmk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32970",
  "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",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214722935",
  "pdf_url": "https://arxiv.org/pdf/2609.32970",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2k1txcraxrhmk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.32970",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32970"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Obtaining Game-Stationary Points for Smooth Nonconvex-Nonconcave Minimax Problems via First-Order Methods",
  "title_text": "Obtaining Game-Stationary Points for Smooth Nonconvex-Nonconcave Minimax Problems via First-Order Methods",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/2k1txcraxrhmk",
  "venue": "arXiv",
  "versions": []
}