{
  "id": "2206.09462",
  "version": "v1",
  "published": "2022-06-19T18:18:21.000Z",
  "updated": "2022-06-19T18:18:21.000Z",
  "title": "Fast Krasnosel'skii-Mann algorithm with a convergence rate of the fixed point iteration of $o(1/k)$",
  "authors": [
    "Radu Ioan Bot",
    "Dang-Khoa Nguyen"
  ],
  "categories": [
    "math.OC",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "The Krasnosel'skii-Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We enhance the Krasnosel'skii-Mann algorithm with Nesterov's momentum updates and show that the resulting numerical method exhibits a convergence rate for the fixed point residual of $o(1/k)$ while preserving the weak convergence of the iterates to a fixed point of the operator. Numerical experiments illustrate the superiority of the resulting so-called Fast KM algorithm over various fixed point iterative schemes, and also its oscillatory behavior, which is a specific of Nesterov's momentum optimization algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-19T18:18:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47J20",
      "47H05",
      "65K15",
      "65Y20"
    ],
    "keywords": [
      "fast krasnoselkii-mann algorithm",
      "fixed point iteration",
      "convergence rate",
      "nesterovs momentum optimization algorithms",
      "iterative scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}