{
  "id": "2105.10607",
  "version": "v1",
  "published": "2021-05-21T23:50:35.000Z",
  "updated": "2021-05-21T23:50:35.000Z",
  "title": "Towards the Biconjugate of Bivariate Piecewise Quadratic Functions",
  "authors": [
    "Deepak Kumar",
    "Yves Lucet"
  ],
  "comment": "11 pages, 3 figures",
  "journal": "In: Le Thi H., Le H., Pham Dinh T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer",
  "doi": "10.1007/978-3-030-21803-4_27",
  "categories": [
    "math.OC"
  ],
  "abstract": "Computing the closed convex envelope or biconjugate is the core operation that bridges the domain of nonconvex with convex analysis. We focus here on computing the conjugate of a bivariate piecewise quadratic function defined over a polytope. First, we compute the convex envelope of each piece, which is characterized by a polyhedral subdivision such that over each member of the subdivision, it has a rational form (square of a linear function over a linear function). Then we compute the conjugate of all such rational functions. It is observed that the conjugate has a parabolic subdivision such that over each member of its subdivision, it has a fractional form (linear function over square root of a linear function). This computation of the conjugate is performed with a worst-case linear time complexity algorithm. Our results are an important step toward computing the conjugate of a piecewise quadratic function, and further in obtaining explicit formulas for the convex envelope of piecewise rational functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-21T23:50:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C26"
    ],
    "keywords": [
      "bivariate piecewise quadratic function",
      "linear function",
      "convex envelope",
      "worst-case linear time complexity algorithm",
      "biconjugate"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}