{
  "id": "2106.12294",
  "version": "v1",
  "published": "2021-06-23T10:22:28.000Z",
  "updated": "2021-06-23T10:22:28.000Z",
  "title": "Improved convergence rates and trajectory convergence for primal-dual dynamical systems with vanishing damping",
  "authors": [
    "Radu Ioan Bot",
    "Dang-Khoa Nguyen"
  ],
  "categories": [
    "math.OC",
    "math.CA"
  ],
  "abstract": "In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with asymptotically vanishing damping term. The system is formulated in terms of the augmented Lagrangian associated to the minimization problem. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. In case the objective function has Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically weakly converges to a primal-dual optimal solution of the underlying minimization problem. To the best of our knowledge, this is the first result which guarantees the convergence of the trajectory generated by a primal-dual dynamical system with asymptotic vanishing damping. Moreover, we will rediscover in case of the unconstrained minimization of a convex differentiable function with Lipschitz continuous gradient all convergence statements obtained in the literature for Nesterov's accelerated gradient method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-23T10:22:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37N40",
      "46N10",
      "65K10",
      "90C25"
    ],
    "keywords": [
      "primal-dual dynamical system",
      "vanishing damping",
      "trajectory convergence",
      "convergence rates",
      "lipschitz continuous gradient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}