{
  "id": "2409.00655",
  "version": "v1",
  "published": "2024-09-01T08:15:11.000Z",
  "updated": "2024-09-01T08:15:11.000Z",
  "title": "The landscape of deterministic and stochastic optimal control problems: One-shot Optimization versus Dynamic Programming",
  "authors": [
    "Jihun Kim",
    "Yuhao Ding",
    "Yingjie Bi",
    "Javad Lavaei"
  ],
  "comment": "16 pages, 4 figures",
  "doi": "10.1109/TAC.2024.3415459",
  "categories": [
    "math.OC"
  ],
  "abstract": "Optimal control problems can be solved via a one-shot (single) optimization or a sequence of optimization using dynamic programming (DP). However, the computation of their global optima often faces NP-hardness, and thus only locally optimal solutions may be obtained at best. In this work, we consider the discrete-time finite-horizon optimal control problem in both deterministic and stochastic cases and study the optimization landscapes associated with two different approaches: one-shot and DP. In the deterministic case, we prove that each local minimizer of the one-shot optimization corresponds to some control input induced by a locally minimum control policy of DP, and vice versa. However, with a parameterized policy approach, we prove that deterministic and stochastic cases both exhibit the desirable property that each local minimizer of DP corresponds to some local minimizer of the one-shot optimization, but the converse does not necessarily hold. Nonetheless, under different technical assumptions for deterministic and stochastic cases, if there exists only a single locally minimum control policy, one-shot and DP turn out to capture the same local solution. These results pave the way to understand the performance and stability of local search methods in optimal control.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-01T08:15:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37N35",
      "49K30",
      "49K45",
      "90C39",
      "93E20"
    ],
    "keywords": [
      "stochastic optimal control problems",
      "one-shot optimization",
      "locally minimum control policy",
      "dynamic programming",
      "finite-horizon optimal control problem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "IEEE"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}