{
  "id": "1910.07873",
  "version": "v1",
  "published": "2019-10-17T13:02:16.000Z",
  "updated": "2019-10-17T13:02:16.000Z",
  "title": "On The Strong Convergence of The Gradient Projection Algorithm with Tikhonov regularizing term",
  "authors": [
    "Ramzi May"
  ],
  "comment": "11 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "We investigate the strong and the weak convergence properties of the following gradient projection algorithm with Tikhonov regularizing term \\[ x_{n+1}=P_{Q}(x_{n}-\\gamma_{n}\\nabla f(x_{n})-\\gamma_{n}\\alpha_{n}\\nabla \\phi (x_{n})), \\] where $P_{Q}$ is the projection operator from a Hilbert space $\\mathcal{H}$ onto a given nonempty, closed and convex subset $Q,$ $f:\\mathcal{H}% \\rightarrow \\mathbb{R}$ a regular convex function, $\\phi :\\mathcal{H}% \\rightarrow \\mathbb{R}$ a regular strongly convex function, and $\\gamma_{n}$ and $\\alpha_{n}$ are positive real numbers. Following a Lyuapunov approach inspired essentially from the paper [Comminetti R, Peypouquet J Sorin S. Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization. J. Differential Equations. (2001); 245:3753-3763], we establish the strong convergence of $(x_{n})_{n}$ to a particular minimizer $x^{\\ast }$ of $f$ on $Q$ under some simple and natural conditions on the objective function $f$\\ and the sequences $(\\gamma_{n})_{n}$ and $(\\alpha_{n})_{n}$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-17T13:02:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46N10",
      "90C25"
    ],
    "keywords": [
      "gradient projection algorithm",
      "tikhonov regularizing term",
      "strong convergence",
      "regular convex function",
      "regular strongly convex function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}