{
  "abstract": "We consider the deterministic oracle complexity of finding a $(δ,ε)$-Goldstein stationary point of a Lipschitz function under constraints. The problem is posed in $\\mathbb{R}^d$, where $d$ denotes the dimension; constraints are specified either by a closed feasible set or by a single explicit inequality. For closed feasible sets that are star-shaped with respect to an interior ball, including all convex bodies, we use a resisting-oracle construction to show that every deterministic algorithm needs $Ω(ε^{-2})$ objective-oracle calls in the worst case when $d\\gtrsimε^{-2}$ and $δ\\lesssimε$. For explicit inequalities satisfying uniform Goldstein constraint qualification and strict feasibility, finding Goldstein KKT stationary points requires exponentially many oracle calls in $d$ in the worst case, even at fixed stationarity accuracy. This exponential dependence holds under dimension-independent constraint regularity and strict-feasibility bounds, with a sufficiently small neighborhood radius and complementary slackness error. The result covers common constraints such as ball and box constraints and also allows nonconvex feasible regions. For convex coercive constraints with a Slater point, the resisting-oracle argument also gives an $Ω(ε^{-2})$ oracle-call bound in the same dimension and radius regime while allowing a larger complementary slackness error. These results hold for all deterministic algorithms using a local oracle, whether or not the returned point was queried. Each constructed objective has a strictly feasible global minimizer; the lower bounds therefore concern the information required to find stationary points, not their existence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06739",
  "author_total": 2,
  "authors": [
    {
      "name": "Yixi Ding"
    },
    {
      "name": "Ya-xiang Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgw4zse15wwgc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06739",
  "fulltext_url": null,
  "id": "dgw4zse15wwgc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06739",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06739",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgw4zse15wwgc.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06739",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06739"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Deterministic Oracle Complexity for Goldstein Stationarity in Nonsmooth Nonconvex Optimization with Constraints",
  "title_text": "Deterministic Oracle Complexity for Goldstein Stationarity in Nonsmooth Nonconvex Optimization with Constraints",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/dgw4zse15wwgc",
  "venue": "arXiv",
  "versions": []
}