{
  "id": "1307.5151",
  "version": "v1",
  "published": "2013-07-19T08:00:00.000Z",
  "updated": "2013-07-19T08:00:00.000Z",
  "title": "SDP Duals without Duality Gaps for a Class of Convex Minimax Programs",
  "authors": [
    "V. Jeyakumar",
    "J. Vicente-Perez"
  ],
  "journal": "J Optim Theory Appl 2013",
  "doi": "10.1007/s10957-013-0496-0",
  "categories": [
    "math.OC"
  ],
  "abstract": "In this paper we introduce a new dual program, which is representable as a semi-definite linear programming problem, for a primal convex minimax programming model problem and show that there is no duality gap between the primal and the dual whenever the functions involved are SOS-convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust SOS-convex programming problems under data uncertainty and to minimax fractional programming problems with SOS-convex polynomials. We obtain these results by first establishing sum of squares polynomial representations of non-negativity of a convex max function over a system of SOS-convex constraints. The new class of SOS-convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The SOS-convexity of polynomials can numerically be checked by solving semi-definite programming problems whereas numerically verifying convexity of polynomials is generally very hard.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-19T08:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex minimax programs",
      "duality gap",
      "sdp duals",
      "minimax programming model problem",
      "programming problem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.5151J"
    }
  }
}