{
  "id": "2109.13467",
  "version": "v2",
  "published": "2021-09-28T03:39:06.000Z",
  "updated": "2023-04-25T08:13:29.000Z",
  "title": "A unified differential equation solver approach for separable convex optimization: splitting, acceleration and nonergodic rate",
  "authors": [
    "Hao Luo",
    "Zihang Zhang"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "This paper provides a self-contained ordinary differential equation solver approach for separable convex optimization problems. A novel primal-dual dynamical system with built-in time rescaling factors is introduced, and the exponential decay of a tailored Lyapunov function is established. Then several time discretizations of the continuous model are considered and analyzed via a unified discrete Lyapunov function. Moreover, two families of accelerated proximal alternating direction methods of multipliers are obtained, and nonergodic optimal mixed-type convergence rates shall be proved for the primal objective residual, the feasibility violation and the Lagrangian gap. Finally, numerical experiments are provided to validate the practical performances.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-04-25T08:13:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37N40",
      "65K05",
      "90C25"
    ],
    "keywords": [
      "unified differential equation solver approach",
      "separable convex optimization",
      "optimal mixed-type convergence rates",
      "nonergodic rate",
      "ordinary differential equation solver"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}