{
  "id": "1807.07554",
  "version": "v1",
  "published": "2018-07-19T17:52:59.000Z",
  "updated": "2018-07-19T17:52:59.000Z",
  "title": "A geometric integration approach to nonsmooth, nonconvex optimisation",
  "authors": [
    "Erlend S. Riis",
    "Matthias J. Ehrhardt",
    "G. R. W. Quispel",
    "Carola-Bibiane Schönlieb"
  ],
  "comment": "33 pages, 11 figures, submitted",
  "categories": [
    "math.OC"
  ],
  "abstract": "The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh--Abe methods. These are generalisations of the Itoh--Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in the smooth setting. We demonstrate that the method and its favourable energy dissipation properties are well-defined in the nonsmooth setting. Furthermore, we prove that whenever the objective function is locally Lipschitz continuous, the iterates almost surely converge to a connected set of Clarke stationary points. We present an implementation of the methods, and apply it to various test problems. The numerical results indicate that the randomised Itoh--Abe methods are superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while remaining competitive in terms of efficiency.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-19T17:52:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49M25",
      "49Q15",
      "65K10",
      "90C15",
      "90C26",
      "90C56",
      "94A08"
    ],
    "keywords": [
      "geometric integration approach",
      "nonconvex optimisation",
      "randomised itoh-abe methods",
      "model parameter optimisation problems",
      "itoh-abe discrete gradient method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}