{
  "id": "2007.12219",
  "version": "v1",
  "published": "2020-07-23T19:16:13.000Z",
  "updated": "2020-07-23T19:16:13.000Z",
  "title": "A First-Order Primal-Dual Method for Nonconvex Constrained Optimization Based On the Augmented Lagrangian",
  "authors": [
    "Daoli Zhu",
    "Lei Zhao",
    "Shuzhong Zhang"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics and data analytics. In this paper, based on the augmented Lagrangian function we introduce a flexible first-order primal-dual method, to be called nonconvex auxiliary problem principle of augmented Lagrangian (NAPP-AL), for solving a class of nonlinearly constrained nonconvex and nonsmooth optimization problems. We demonstrate that NAPP-AL converges to a stationary solution at the rate of o(1/\\sqrt{k}), where k is the number of iterations. Moreover, under an additional error bound condition (to be called VP-EB in the paper), we further show that the convergence rate is in fact linear. Finally, we show that the famous Kurdyka- Lojasiewicz property and the metric subregularity imply the afore-mentioned VP-EB condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-23T19:16:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C30",
      "90C26"
    ],
    "keywords": [
      "nonconvex constrained optimization",
      "augmented lagrangian",
      "nonlinearly constrained nonconvex",
      "nonconvex auxiliary problem principle",
      "nonsmooth optimization models play"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}