A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866916230830489600 |
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| author | Reisinger, Christoph Stockinger, Wolfgang Zhang, Yufei |
| author_facet | Reisinger, Christoph Stockinger, Wolfgang Zhang, Yufei |
| contents | We propose a PDE-based accelerated gradient algorithm for optimal feedback controls of McKean-Vlasov dynamics that involve mean-field interactions both in the state and action. The method exploits a forward-backward splitting approach and iteratively refines the approximate controls based on the gradients of smooth costs, the proximal maps of nonsmooth costs, and dynamically updated momentum parameters. At each step, the state dynamics is approximated via a particle system, and the required gradient is evaluated through a coupled system of nonlocal linear PDEs. The latter is solved by finite difference approximation or neural network-based residual approximation, depending on the state dimension. We present exhaustive numerical experiments for low and high-dimensional mean-field control problems, including sparse stabilization of stochastic Cucker-Smale models, which reveal that our algorithm captures important structures of the optimal feedback control and achieves a robust performance with respect to parameter perturbation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_06740 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems Reisinger, Christoph Stockinger, Wolfgang Zhang, Yufei Optimization and Control 49N80, 60H35, 35Q93, 93A16 We propose a PDE-based accelerated gradient algorithm for optimal feedback controls of McKean-Vlasov dynamics that involve mean-field interactions both in the state and action. The method exploits a forward-backward splitting approach and iteratively refines the approximate controls based on the gradients of smooth costs, the proximal maps of nonsmooth costs, and dynamically updated momentum parameters. At each step, the state dynamics is approximated via a particle system, and the required gradient is evaluated through a coupled system of nonlocal linear PDEs. The latter is solved by finite difference approximation or neural network-based residual approximation, depending on the state dimension. We present exhaustive numerical experiments for low and high-dimensional mean-field control problems, including sparse stabilization of stochastic Cucker-Smale models, which reveal that our algorithm captures important structures of the optimal feedback control and achieves a robust performance with respect to parameter perturbation. |
| title | A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems |
| topic | Optimization and Control 49N80, 60H35, 35Q93, 93A16 |
| url | https://arxiv.org/abs/2108.06740 |