{
  "abstract": "We study deterministic first-order bilevel optimization under weak lower-level convexity, allowing nonconvex lower-level objectives and without assuming strong convexity, the Polyak-Łojasiewicz condition, or an error-bound property. We consider a $δ$-relaxed Moreau-gap constraint, with $δ>0$, for the lower-level stationarity condition and propose an inexact variable-smoothing penalty method (IVSP) for computing its approximate Karush-Kuhn-Tucker (KKT) points. For any fixed relaxation level $δ$, under a standard extended no-nonzero-abnormal-multiplier constraint qualification (ENNAMCQ), we prove finite stabilization of the adaptive penalty parameter and an overall $\\widetilde O(\\varepsilon^{-3})$ first-order complexity for computing an $\\varepsilon$-KKT point. Notably, we give verifiable sufficient conditions for ENNAMCQ covering convex lower-level objectives without nonconstant affine segments (including the strictly convex case), and a class of nonconvex sample-reweighting models. The positive relaxation avoids the intrinsic constraint-qualification degeneracy of the exact Moreau-gap constraint while achieving an $\\mathcal O(\\sqrtδ)$ lower-level near-stationarity guarantee. Numerical experiments on synthetic and real-world bilevel learning problems illustrate the practical performance of IVSP.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39736",
  "author_total": 3,
  "authors": [
    {
      "name": "Jan Harold Alcantara"
    },
    {
      "name": "Masahiro Inoue"
    },
    {
      "name": "Akiko Takeda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ar5x3tqgxr66p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39736",
  "fulltext_url": null,
  "id": "ar5x3tqgxr66p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39736",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7215074586",
  "pdf_url": "https://arxiv.org/pdf/2609.39736",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ar5x3tqgxr66p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39736",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39736"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Improved KKT Complexity for First-Order Bilevel Optimization under Weak Lower-Level Convexity",
  "title_text": "Improved KKT Complexity for First-Order Bilevel Optimization under Weak Lower-Level Convexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/ar5x3tqgxr66p",
  "venue": "arXiv",
  "versions": []
}