{
  "id": "2409.13118",
  "version": "v1",
  "published": "2024-09-19T23:03:54.000Z",
  "updated": "2024-09-19T23:03:54.000Z",
  "title": "Lipschitz stability of least-squares problems regularized by functions with $\\mathcal{C}^2$-cone reducible conjugates",
  "authors": [
    "Ying Cui",
    "Tim Hoheisel",
    "Tran T. A. Nghia",
    "Defeng Sun"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "In this paper, we study Lipschitz continuity of the solution mappings of regularized least-squares problems for which the convex regularizers have (Fenchel) conjugates that are $\\mathcal{C}^2$-cone reducible. Our approach, by using Robinson's strong regularity on the dual problem, allows us to obtain new characterizations of Lipschitz stability that rely solely on first-order information, thus bypassing the need to explore second-order information (curvature) of the regularizer. We show that these solution mappings are automatically Lipschitz continuous around the points in question whenever they are locally single-valued. We leverage our findings to obtain new characterizations of full stability and tilt stability for a broader class of convex additive-composite problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-19T23:03:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J52",
      "49J53",
      "49K40",
      "90C25",
      "90C31"
    ],
    "keywords": [
      "cone reducible conjugates",
      "lipschitz stability",
      "solution mappings",
      "convex additive-composite problems",
      "study lipschitz continuity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}