Integer Optimal Control with Fractional Perimeter Regularization
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914829837533184 |
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| author | Antil, Harbir Manns, Paul |
| author_facet | Antil, Harbir Manns, Paul |
| contents | Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension $d - 1$ for a domain of dimension $d$.
This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent $0<α<1$. In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a $Γ$-convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent $α$ tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for $α\in (0.5,1)$ under an additional assumption on the gradient of the reduced objective.
The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integer Optimal Control with Fractional Perimeter Regularization Antil, Harbir Manns, Paul Optimization and Control 49M05, 49M25, 49Q20 Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension $d - 1$ for a domain of dimension $d$. This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent $0<α<1$. In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a $Γ$-convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent $α$ tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for $α\in (0.5,1)$ under an additional assumption on the gradient of the reduced objective. The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional. |
| title | Integer Optimal Control with Fractional Perimeter Regularization |
| topic | Optimization and Control 49M05, 49M25, 49Q20 |
| url | https://arxiv.org/abs/2404.04938 |