{
  "id": "2007.12486",
  "version": "v1",
  "published": "2020-07-23T11:09:13.000Z",
  "updated": "2020-07-23T11:09:13.000Z",
  "title": "Regularity and stability for a convex feasibility problem",
  "authors": [
    "Enrico Miglierina",
    "Carlo A. De Bernardi"
  ],
  "comment": "16 pages. arXiv admin note: text overlap with arXiv:1907.13402",
  "categories": [
    "math.OC"
  ],
  "abstract": "Let us consider two sequences of closed convex sets $\\{A_n\\}$ and $\\{B_n\\}$ converging with respect to the Attouch-Wets convergence to $A$ and $B$, respectively. Given a starting point $a_0$, we consider the sequences of points obtained by projecting on the \"perturbed\" sets, i.e., the sequences $\\{a_n\\}$ and $\\{b_n\\}$ defined inductively by $b_n=P_{B_n}(a_{n-1})$ and $a_n=P_{A_n}(b_n)$. Suppose that $A\\cap B$ (or a suitable substitute if $A \\cap B=\\emptyset$) is bounded, we prove that if the couple $(A,B)$ is (boundedly) regular then the couple $(A,B)$ is $d$-stable, i.e., for each $\\{a_n\\}$ and $\\{b_n\\}$ as above we have $\\mathrm{dist}(a_n,A\\cap B)\\to 0$ and $\\mathrm{dist}(b_n,A\\cap B)\\to 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-23T11:09:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47J25",
      "90C25",
      "90C48"
    ],
    "keywords": [
      "convex feasibility problem",
      "regularity",
      "closed convex sets",
      "attouch-wets convergence",
      "starting point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}