{
  "abstract": "We study nonconvex stochastic functions that admit a structured second-order representation, a generalization of lower- and upper-$C^2$ functions to normal integrands. Similarly to the deterministic case, lower-$C^2$ normal integrands are defined as the pointwise maxima over a compact random index set of functions that are twice-continuously differentiable in the decision variable with an integrable bounded Hessian. Our main result is a local value function characterization of this class of functions, namely, that they satisfy a local sufficient ascent property almost surely. Equivalently, they differ from a convex normal integrand by a random quadratic. The upper-$C^2$ theory follows by a change of sign, and delivers the corresponding lower envelopes, the form in which these integrands appear as second-stage value functions of stochastic programs with compact recourse. We also show that for this class integrability of the function values, of the local Lipschitz moduli, and of the minimum-norm Clarke subgradients are equivalent. We then study smooth integrands, finite lower envelopes, difference-of-convex integrands, Moreau and forward--backward envelopes, compositions with a smooth map, monotone convex compositions, and compact-recourse stochastic programs within these classes of nonconvex stochastic functions. As an application, we show that the optimistic loss of partial-label regression and the variational formulation of the conditional value-at-risk fit within this framework.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37629",
  "author_total": 4,
  "authors": [
    {
      "name": "Felipe Atenas"
    },
    {
      "name": "Alejandro Jofré"
    },
    {
      "name": "Pedro Pérez-Aros"
    },
    {
      "name": "David Torregrosa-Belén"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3kmmnmzkckb7s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37629",
  "fulltext_url": null,
  "id": "3kmmnmzkckb7s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37629",
  "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": "W7224086232",
  "pdf_url": "https://arxiv.org/pdf/2610.37629",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3kmmnmzkckb7s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37629",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37629"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Lower- and upper-$C^2$ integrands: a structured class of functions for stochastic nonconvex optimization",
  "title_text": "Lower- and upper-C^2 integrands: a structured class of functions for stochastic nonconvex optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/3kmmnmzkckb7s",
  "venue": "arXiv",
  "versions": []
}