{
  "id": "2206.07204",
  "version": "v1",
  "published": "2022-06-14T23:09:27.000Z",
  "updated": "2022-06-14T23:09:27.000Z",
  "title": "The range of the Douglas-Rachford operator in infinite-dimensional Hilbert spaces",
  "authors": [
    "Walaa M. Moursi"
  ],
  "categories": [
    "math.OC",
    "math.FA"
  ],
  "abstract": "The Douglas-Rachford algorithm is one of the most prominent splitting algorithms for solving convex optimization problems. Recently, the method has been successful in finding a generalized solution (provided that one exists) for optimization problems in the inconsistent case, i.e., when a solution does not exist. The convergence analysis of the inconsistent case hinges on the study of the range of the displacement operator associated with the Douglas-Rachford splitting operator and the corresponding minimal displacement vector. In this paper, we provide a formula for the range of the Douglas-Rachford splitting operator in (possibly) infinite-dimensional Hilbert space under mild assumptions on the underlying operators. Our new results complement known results in finite-dimensional Hilbert spaces. Several examples illustrate and tighten our conclusions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-14T23:09:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47H05",
      "47H09",
      "90C25",
      "90C46",
      "47H14",
      "49M27",
      "49M29",
      "49N15"
    ],
    "keywords": [
      "infinite-dimensional hilbert space",
      "douglas-rachford operator",
      "douglas-rachford splitting operator",
      "corresponding minimal displacement vector",
      "finite-dimensional hilbert spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}