{
  "abstract": "We assemble a finite-dimensional \\(\\mathcal{VU}\\)-calculus for composite nonsmooth functions whose outer function is convex: the chain rule, separable and convex sums, a strictly differentiable perturbation of a lower semicontinuous (lsc) term, the model \\(δ_X+f_0+θ\\circ F\\), and a finite maximum of \\(C^1\\) functions. The same algebra yields an \\(\\varepsilon\\)-\\(\\mathcal{VU}\\) chain rule for a proper outer approximation of the subdifferential. On the set \\(\\mathcal{R}_{h,F}\\) of points at which the convex chain rule holds, the subspaces \\(\\mathcal{V}f\\) and \\(\\mathcal{U}f\\), an orthonormal frame of \\(\\mathcal{U}f\\), and the \\(\\mathcal{U}\\)-gradient \\(\\bar g_u\\) are written in terms of the factors. If \\(f\\) is \\(C^1\\)-partly smooth at \\(\\bar{x}\\), that gradient is \\(\\bar g_u=U_f^\\top\\nabla_{\\mathcal{M}}f(\\bar{x})\\). If \\(f\\) is \\(C^2\\)-partly smooth and \\(0\\in\\ri\\partial f(\\bar{x})\\), the convex \\(\\mathcal{U}\\)-Hessian \\(H_U\\) (when \\(f\\) is convex) or the local matrix \\(H_\\varepsilon\\) (when \\(f\\) is prox-regular at \\(\\bar{x}\\) for \\(0\\)) equals the Gram matrix \\(U_f^\\top\\nabla^2_{\\mathcal{M}}f(\\bar{x})\\,U_f\\); the same matrix, written along the active manifold in a continuous frame, depends continuously on the base point. If in addition \\(f\\) is \\(C^2\\)-partly smooth at \\(\\bar{x}\\), then under prox-regularity and subdifferential continuity at \\(\\bar{x}\\) for \\(0\\), \\(H_\\varepsilon\\succ 0\\) is equivalent to tilt stability of \\(\\bar{x}\\) and to strong metric regularity of \\(\\partial f\\) at \\((\\bar{x},0)\\), with \\(\\lip\\bigl((\\partial f)^{-1}\\bigr)(0\\mid\\bar{x})=\\|H_\\varepsilon^{-1}\\|\\). The calculus and the test are illustrated on a hinge composite and two elementary tests of \\(H_U\\succ 0\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18683",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuai Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3m5xwg6twgmde.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18683",
  "fulltext_url": null,
  "id": "3m5xwg6twgmde",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18683",
  "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": "W7213523782",
  "pdf_url": "https://arxiv.org/pdf/2609.18683",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3m5xwg6twgmde.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18683",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18683"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A $\\mathcal{VU}$-calculus for composite functions and the $\\mathcal{U}$-Hessian of partly smooth functions",
  "title_text": "A 𝒱𝒰-calculus for composite functions and the 𝒰-Hessian of partly smooth functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/3m5xwg6twgmde",
  "venue": "arXiv",
  "versions": []
}