{
  "id": "1905.08142",
  "version": "v1",
  "published": "2019-05-20T14:43:06.000Z",
  "updated": "2019-05-20T14:43:06.000Z",
  "title": "Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets",
  "authors": [
    "Lucas Slot",
    "Monique Laurent"
  ],
  "comment": "27 pages with 6 figures",
  "categories": [
    "math.OC"
  ],
  "abstract": "We consider the problem of computing the minimum value $f_{\\min,K}$ of a polynomial $f$ over a compact set $K \\subseteq \\mathbb{R}^n$, which can be reformulated as finding a probability measure $\\nu$ on $K$ minimizing $\\int_K f d\\nu$. Lasserre showed that it suffices to consider such measures of the form $\\nu = q\\mu$, where $q$ is a sum-of-squares polynomial and $\\mu$ is a given Borel measure supported on $K$. By bounding the degree of $q$ by $2r$ one gets a converging hierarchy of upper bounds $f^{(r)}$ for $f_{\\min,K}$. When $K$ is the hypercube $[-1, 1]^n$, equipped with the Chebyshev measure, the parameters $f^{(r)}$ are known to converge to $f_{\\min,K}$ at a rate in $O(1/r^2)$. We extend this error estimate to a wider class of convex bodies, while also allowing for a broader class of reference measures, including the Lebesgue measure. Our analysis applies to simplices, balls and convex bodies that locally look like a ball. In addition, we show an error estimate in $O(\\log r / r)$ when $K$ satisfies a minor geometrical condition, and in $O(\\log^2 r / r^2)$ when $K$ is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in $O(1 / \\sqrt{r})$ and $O(1/r)$ for these two respective cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-20T14:43:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C22",
      "90C26",
      "90C30"
    ],
    "keywords": [
      "lasserres measure-based upper bounds",
      "compact set",
      "polynomial minimization",
      "convergence analysis",
      "error estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}