Rockafellian Relaxation for PDE-Constrained Optimization with Distributional Uncertainty
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915662784364544 |
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| author | Antil, Harbir Carney, Sean P. Díaz, Hugo Royset, Johannes O. |
| author_facet | Antil, Harbir Carney, Sean P. Díaz, Hugo Royset, Johannes O. |
| contents | Stochastic optimization problems are generally known to be ill-conditioned to the form of the underlying uncertainty. A framework is introduced for optimal control problems with partial differential equations as constraints that is robust to inaccuracies in the precise form of the problem uncertainty. The framework is based on problem relaxation and involves optimizing a bivariate, "Rockafellian" objective functional that features both a standard control variable and an additional perturbation variable that handles the distributional ambiguity. In the presence of distributional corruption, the Rockafellian objective functionals are shown in the appropriate settings to $Γ$-converge to uncorrupted objective functionals in the limit of vanishing corruption. Numerical examples illustrate the framework's utility for outlier detection and removal and for variance reduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rockafellian Relaxation for PDE-Constrained Optimization with Distributional Uncertainty Antil, Harbir Carney, Sean P. Díaz, Hugo Royset, Johannes O. Optimization and Control 49M37, 90C30, 93C20, 93E20, 49K20, 49J20 Stochastic optimization problems are generally known to be ill-conditioned to the form of the underlying uncertainty. A framework is introduced for optimal control problems with partial differential equations as constraints that is robust to inaccuracies in the precise form of the problem uncertainty. The framework is based on problem relaxation and involves optimizing a bivariate, "Rockafellian" objective functional that features both a standard control variable and an additional perturbation variable that handles the distributional ambiguity. In the presence of distributional corruption, the Rockafellian objective functionals are shown in the appropriate settings to $Γ$-converge to uncorrupted objective functionals in the limit of vanishing corruption. Numerical examples illustrate the framework's utility for outlier detection and removal and for variance reduction. |
| title | Rockafellian Relaxation for PDE-Constrained Optimization with Distributional Uncertainty |
| topic | Optimization and Control 49M37, 90C30, 93C20, 93E20, 49K20, 49J20 |
| url | https://arxiv.org/abs/2405.00176 |