{
  "id": "2212.12256",
  "version": "v1",
  "published": "2022-12-23T11:08:01.000Z",
  "updated": "2022-12-23T11:08:01.000Z",
  "title": "On a fixed-point continuation method for a convex optimization problem",
  "authors": [
    "Jean-Baptiste Fest",
    "Tommi Heikkilä",
    "Ignace Loris",
    "Ségolène Martin",
    "Luca Ratti",
    "Simone Rebegoldi",
    "Gesa Sarnighausen"
  ],
  "comment": "15 pages, 2 figures. Workshop on Advanced Techniques in Optimization for Machine learning and Imaging",
  "categories": [
    "math.OC",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "We consider a variation of the classical proximal-gradient algorithm for the iterative minimization of a cost function consisting of a sum of two terms, one smooth and the other prox-simple, and whose relative weight is determined by a penalty parameter. This so-called fixed-point continuation method allows one to approximate the problem's trade-off curve, i.e. to compute the minimizers of the cost function for a whole range of values of the penalty parameter at once. The algorithm is shown to converge, and a rate of convergence of the cost function is also derived. Furthermore, it is shown that this method is related to iterative algorithms constructed on the basis of the $\\epsilon$-subdifferential of the prox-simple term. Some numerical examples are provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-23T11:08:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fixed-point continuation method",
      "convex optimization problem",
      "cost function",
      "penalty parameter",
      "problems trade-off curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}