{
  "id": "1703.01339",
  "version": "v1",
  "published": "2017-03-03T21:03:09.000Z",
  "updated": "2017-03-03T21:03:09.000Z",
  "title": "Newton-like dynamics associated to nonconvex optimization problems",
  "authors": [
    "Radu Ioan Bot",
    "Ernö Robert Csetnek"
  ],
  "categories": [
    "math.OC",
    "math.DS"
  ],
  "abstract": "We consider the dynamical system \\begin{equation*}\\left\\{ \\begin{array}{ll} v(t)\\in\\partial\\phi(x(t))\\\\ \\lambda\\dot x(t) + \\dot v(t) + v(t) + \\nabla \\psi(x(t))=0, \\end{array}\\right.\\end{equation*} where $\\phi:\\R^n\\to\\R\\cup\\{+\\infty\\}$ is a proper, convex and lower semicontinuous function, $\\psi:\\R^n\\to\\R$ is a (possibly nonconvex) smooth function and $\\lambda>0$ is a parameter which controls the velocity. We show that the set of limit points of the trajectory $x$ is contained in the set of critical points of the objective function $\\phi+\\psi$, which is here seen as the set of the zeros of its limiting subdifferential. If the objective function satisfies the Kurdyka-\\L{}ojasiewicz property, then we can prove convergence of the whole trajectory $x$ to a critical point. Furthermore, convergence rates for the orbits are obtained in terms of the \\L{}ojasiewicz exponent of the objective function, provided the latter satisfies the \\L{}ojasiewicz property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-03T21:03:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34G25",
      "47J25",
      "47H05",
      "90C26",
      "90C30",
      "65K10"
    ],
    "keywords": [
      "nonconvex optimization problems",
      "newton-like dynamics",
      "critical point",
      "limit points",
      "trajectory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}