{
  "id": "1311.2367",
  "version": "v1",
  "published": "2013-11-11T07:37:09.000Z",
  "updated": "2013-11-11T07:37:09.000Z",
  "title": "Potentialities of Nonsmooth Optimization",
  "authors": [
    "Vsevolod Ivanov Ivanov"
  ],
  "comment": "25 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "In this paper, we show that higher-order optimality conditions can be obtain for arbitrary nonsmooth function. We introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is consistent with the classical higher-order Fr\\'echet directional derivative in the sense that both derivatives of the same order coincide if the last one exists. We obtain necessary and sufficient conditions of order $n$ ($n$ is a positive integer) for a local minimum and isolated local minimum of order $n$ in terms of these derivatives and subdifferentials. We do not require any restrictions on the function in our results. A special class $\\mathcal F_n$ of functions is defined and optimality conditions for isolated local minimum of order $n$ for a function $f\\in\\mathcal F_n$ are derived. The derivative of order $n$ does not appear in these characterizations. We prove necessary and sufficient criteria such that every stationary point of order $n$ is a global minimizer. We compare our results with some previous ones.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-11T07:37:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49K10",
      "90C46",
      "26B05",
      "26B25"
    ],
    "keywords": [
      "nonsmooth optimization",
      "higher-order frechet directional derivative",
      "isolated local minimum",
      "potentialities",
      "classical higher-order frechet directional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.2367I"
    }
  }
}