{
  "id": "1904.08828",
  "version": "v1",
  "published": "2019-04-18T15:10:23.000Z",
  "updated": "2019-04-18T15:10:23.000Z",
  "title": "Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere",
  "authors": [
    "Etienne de Klerk",
    "Monique Laurent"
  ],
  "comment": "14 pages, 2 figures",
  "categories": [
    "math.OC"
  ],
  "abstract": "We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the exact rate of convergence is Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-18T15:10:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C22",
      "90C26",
      "90C30"
    ],
    "keywords": [
      "upper bound",
      "lasserre hierarchy",
      "convergence analysis",
      "polynomial minimization problems",
      "probability density function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}