{
  "id": "1009.0125",
  "version": "v3",
  "published": "2010-09-01T09:22:13.000Z",
  "updated": "2011-05-12T13:12:57.000Z",
  "title": "A new look at nonnegativity on closed sets and polynomial optimization",
  "authors": [
    "Jean B. Lasserre"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "We first show that a continuous function f is nonnegative on a closed set $K\\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\\nu =fd\\mu$ with support equal to K, are all positive semidefinite (if $K$ is compact $\\mu$ is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with {\\it no} lifting, of the cone of nonnegative polynomials of degree at most $d$. Wen used in polynomial optimization on certain simple closed sets $\\K$ (like e.g., the whole space $\\R^n$, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-05-12T13:12:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "closed set",
      "polynomial optimization",
      "semidefinite",
      "nonnegativity",
      "convergent sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.0125L"
    }
  }
}