{
  "abstract": "We introduce a tensor envelope (TEE) for composite objectives $φ=f+g$, built from a $q$th-order Taylor model of the smooth term $f$ regularized by a $p$th power with $p>q$. To construct it, we first establish a tensor descent lemma and characterize its behavior in the three regimes $q<p<q+1$, $p=q+1$, and $p>q+1$. Under suitable prox-boundedness and parameter conditions, the associated tensor operator (TOP) has nonempty compact values, and TEE is finite-valued and continuous, preserves global minimizers, and inherits the level-boundedness of $φ$. We further show that TEE is locally Lipschitz continuous and directionally differentiable, characterize its Fréchet differentiability, and prove that it is $C^1$ whenever TOP is single-valued. As such, TEE preserves the essential optimization landscape properties of $φ$ while enjoying a richer analytical structure, which facilitates the design and analysis of tensor-based optimization methods. Finally, TEE serves as a Lyapunov function for the exact tensor iteration: every cluster point is critical, and, for bounded trajectories with $q<p\\le q+1$, an $ε$-stationary point is reached within $\\mathcal{O}(ε^{-p/(p-1)})$ iterations. To our knowledge, both the tensor descent lemma and this complexity bound are the first of their kind obtained without assuming global Lipschitz or Hölder continuity of the $q$th-order derivative of $f$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04610",
  "author_total": 2,
  "authors": [
    {
      "name": "Alireza Kabgani"
    },
    {
      "name": "Masoud Ahookhosh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/09h1nqs2xr48n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04610",
  "fulltext_url": null,
  "id": "09h1nqs2xr48n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04610",
  "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": "W7220692067",
  "pdf_url": "https://arxiv.org/pdf/2610.04610",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/09h1nqs2xr48n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04610",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04610"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Tensor Envelopes in the Nonconvex Setting",
  "title_text": "Tensor Envelopes in the Nonconvex Setting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/09h1nqs2xr48n",
  "venue": "arXiv",
  "versions": []
}