{
  "id": "1901.01698",
  "version": "v1",
  "published": "2019-01-07T08:17:52.000Z",
  "updated": "2019-01-07T08:17:52.000Z",
  "title": "Local minimizers of semi-algebraic functions",
  "authors": [
    "Tien-Son Pham"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "Consider a semi-algebraic function $f\\colon\\mathbb{R}^n \\to {\\mathbb{R}},$ which is continuous around a point $\\bar{x} \\in \\mathbb{R}^n.$ Using the so--called {\\em tangency variety} of $f$ at $\\bar{x},$ we first provide necessary and sufficient conditions for $\\bar{x}$ to be a local minimizer of $f,$ and then in the case where $\\bar{x}$ is an isolated local minimizer of $f,$ we define a ``tangency exponent'' $\\alpha_* > 0$ so that for any $\\alpha \\in \\mathbb{R}$ the following four conditions are always equivalent: (i) the inequality $\\alpha \\ge \\alpha_*$ holds; (ii) the point $\\bar{x}$ is an $\\alpha$-order sharp local minimizer of $f;$ (iii) the limiting subdifferential $\\partial f$ of $f$ is $(\\alpha - 1)$-order strongly metrically subregular at $\\bar{x}$ for $0;$ and (iv) the function $f$ satisfies the \\L ojaseiwcz gradient inequality at $\\bar{x}$ with the exponent $1 - \\frac{1}{\\alpha}.$ Besides, we also present a counterexample to a conjecture posed by Drusvyatskiy and Ioffe in [Math. Program. Ser. A, 153(2):635--653, 2015].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-07T08:17:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J53",
      "14P10",
      "65K10",
      "49J52",
      "26D10"
    ],
    "keywords": [
      "semi-algebraic function",
      "order sharp local minimizer",
      "ojaseiwcz gradient inequality",
      "order strongly metrically subregular",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}