{
  "id": "1906.05608",
  "version": "v1",
  "published": "2019-06-13T11:37:40.000Z",
  "updated": "2019-06-13T11:37:40.000Z",
  "title": "Non-convex optimization via strongly convex majoirziation-minimization",
  "authors": [
    "Azita Mayeli"
  ],
  "comment": "9 pages, 1 figure",
  "categories": [
    "math.OC",
    "cs.NA",
    "math.CA",
    "math.NA"
  ],
  "abstract": "In this paper, we introduce a class of nonsmooth nonconvex least square optimization problem using convex analysis tools and we propose to use the iterative minimization-majorization (MM) algorithm on a convex set with initializer away from the origin to find an optimal point for the optimization problem. For this, first we use an approach to construct a class of convex majorizers which approximate the value of non-convex cost function on a convex set. The convergence of the iterative algorithm is guaranteed when the initial point $x^{(0)}$ is away from the origin and the iterative points $x^{(k)}$ are obtained in a ball centred at $x^{(k-1)}$ with small radius. The algorithm converges to a stationary point of cost function when the surregators are strongly convex. For the class of our optimization problems, the proposed penalizer of the cost function is the difference of $\\ell_1$-norm and the Moreau envelope of a convex function, and it is a generalization of GMC non-separable penalty function previously introduced by Ivan Selesnick in \\cite{IS17}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-13T11:37:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65K05",
      "65K10",
      "26B25",
      "90C26",
      "90C30"
    ],
    "keywords": [
      "strongly convex majoirziation-minimization",
      "non-convex optimization",
      "convex set",
      "convex analysis tools",
      "gmc non-separable penalty function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}