{
  "id": "1507.06143",
  "version": "v1",
  "published": "2015-07-22T12:07:59.000Z",
  "updated": "2015-07-22T12:07:59.000Z",
  "title": "Semidefinite approximations of projections and polynomial images of semialgebraic sets",
  "authors": [
    "Victor Magron",
    "Didier Henrion",
    "Jean-Bernard Lasserre"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "Given a compact semialgebraic set S of R^n and a polynomial map f from R^n to R^m, we consider the problem of approximating the image set F = f(S) in R^m. This includes in particular the projection of S on R^m for n greater than m. Assuming that F is included in a set B which is \"simple\" (e.g. a box or a ball), we provide two methods to compute certified outer approximations of F. Method 1 exploits the fact that F can be defined with an existential quantifier, while Method 2 computes approximations of the support of image measures.The two methods output a sequence of superlevel sets defined with a single polynomial that yield explicit outer approximations of F. Finding the coefficients of this polynomial boils down to computing an optimal solution of a convex semidefinite program. We provide guarantees of strong convergence to F in L^1 norm on B, when the degree of the polynomial approximation tends to infinity. Several examples of applications are provided, together with numerical experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-22T12:07:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semidefinite approximations",
      "polynomial images",
      "projection",
      "yield explicit outer approximations",
      "polynomial approximation tends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150706143M"
    }
  }
}