{
  "id": "2107.12122",
  "version": "v1",
  "published": "2021-07-26T11:40:43.000Z",
  "updated": "2021-07-26T11:40:43.000Z",
  "title": "A Steepest Descent Method for Set Optimization Problems with Set-Valued Mappings of Finite Cardinality",
  "authors": [
    "Gemayqzel Bouza",
    "Ernest Quintana",
    "Christiane Tammer"
  ],
  "journal": "J Optim Theory Appl (2021)",
  "doi": "10.1007/s10957-021-01887-y",
  "categories": [
    "math.OC"
  ],
  "abstract": "In this paper, we study a first order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems, and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the PhD thesis by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universit\\\"at Halle-Wittenberg, 2020).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-26T11:40:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C29",
      "90C46",
      "90C47"
    ],
    "keywords": [
      "steepest descent method",
      "finite cardinality",
      "set-valued mappings",
      "optimality conditions",
      "first order solution method"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}