The landscape of deterministic and stochastic optimal control problems: One-shot Optimization versus Dynamic Programming
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914932018118656 |
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| author | Kim, Jihun Ding, Yuhao Bi, Yingjie Lavaei, Javad |
| author_facet | Kim, Jihun Ding, Yuhao Bi, Yingjie Lavaei, Javad |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00655 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The landscape of deterministic and stochastic optimal control problems: One-shot Optimization versus Dynamic Programming Kim, Jihun Ding, Yuhao Bi, Yingjie Lavaei, Javad Optimization and Control 37N35, 49K30, 49K45, 90C39, 93E20 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. |
| title | The landscape of deterministic and stochastic optimal control problems: One-shot Optimization versus Dynamic Programming |
| topic | Optimization and Control 37N35, 49K30, 49K45, 90C39, 93E20 |
| url | https://arxiv.org/abs/2409.00655 |