{
  "id": "2001.07999",
  "version": "v1",
  "published": "2020-01-22T13:20:54.000Z",
  "updated": "2020-01-22T13:20:54.000Z",
  "title": "Curiosities and counterexamples in smooth convex optimization",
  "authors": [
    "Jerome Bolte",
    "Edouard Pauwels"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. We show that block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy's gradient curves, convergence of Newton's flow, finite length of Tikhonov path, convergence of central paths, or smooth Kurdyka-Lojasiewicz inequality. All examples are planar. These examples are based on general smooth convex interpolation results. Given a decreasing sequence of positively curved C k convex compact sets in the plane, we provide a level set interpolation of a C k smooth convex function where k $\\ge$ 2 is arbitrary. If the intersection is reduced to one point our interpolant has positive definite Hessian, otherwise it is positive definite out of the solution set. Furthermore , given a sequence of decreasing polygons we provide an interpolant agreeing with the vertices and whose gradients coincide with prescribed normals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-22T13:20:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smooth convex optimization",
      "counterexamples",
      "general smooth convex interpolation results",
      "curiosities",
      "smooth kurdyka-lojasiewicz inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}