{
  "id": "1209.3049",
  "version": "v1",
  "published": "2012-09-13T21:59:19.000Z",
  "updated": "2012-09-13T21:59:19.000Z",
  "title": "Lower bounds on the global minimum of a polynomial",
  "authors": [
    "Mehdi Ghasemi",
    "Jean Bernard Lasserre",
    "Murray Marshall"
  ],
  "journal": "Comput. Optim. Appl. 56(1) (2013)",
  "doi": "10.1007/s10589-013-9596-x",
  "categories": [
    "math.OC",
    "math.AG"
  ],
  "abstract": "We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\\rm gp},M}$ for a multivariate polynomial $f(x) \\in \\mathbb{R}[x]$ of degree $ \\le 2d$ in $n$ variables $x = (x_1,...,x_n)$ on the closed ball ${x \\in \\mathbb{R}^n : \\sum x_i^{2d} \\le M}$, computable by geometric programming, for any real $M$. We compare this bound with the (global) lower bound $f_{{\\rm gp}}$ obtained by Ghasemi and Marshall, and also with the hierarchy of lower bounds, computable by semidefinite programming, obtained by Lasserre [SIAM J. Opt. 11(3) (2001) pp 796-816]. Our computations show that the bound $f_{{\\rm gp},M}$ improves on the bound $f_{{\\rm gp}}$ and that the computation of $f_{{\\rm gp},M}$, like that of $f_{{\\rm gp}}$, can be carried out quickly and easily for polynomials having of large number of variables and/or large degree, assuming a reasonable sparsity of coefficients, cases where the corresponding computation using semidefinite programming breaks down.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-13T21:59:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14P99",
      "65K10",
      "90C25"
    ],
    "keywords": [
      "lower bound",
      "global minimum",
      "computation",
      "semidefinite programming breaks",
      "large number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.3049G"
    }
  }
}