Bouligand Analysis and Discrete Optimal Control of Total Variation-Based Variational Inequalities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912324629037056 |
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| author | Reyes, Juan Carlos De Los |
| author_facet | Reyes, Juan Carlos De Los |
| contents | We investigate differentiability and subdifferentiability properties of the solution mapping associated with variational inequalities (VI) of the second kind involving the discrete total-variation. Bouligand differentiability of the solution operator is established via a direct quotient analysis applied to a primal-dual reformulation of the VI. By exploiting the structure of the directional derivative and introducing a suitable subspace, we fully characterize the Bouligand subdifferential of the solution mapping. We then derive optimality conditions characterizing Bouligand-stationary and strongly-stationary points for discrete VI-constrained optimal control problems. A trust-region algorithm for solving these control problems is proposed based on the obtained characterizations, and a numerical experiment is presented to illustrate the main properties of both the solution and the proposed algorithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_09272 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bouligand Analysis and Discrete Optimal Control of Total Variation-Based Variational Inequalities Reyes, Juan Carlos De Los Optimization and Control 49J40, 49K20, 49M29, 49J52, 90C33 We investigate differentiability and subdifferentiability properties of the solution mapping associated with variational inequalities (VI) of the second kind involving the discrete total-variation. Bouligand differentiability of the solution operator is established via a direct quotient analysis applied to a primal-dual reformulation of the VI. By exploiting the structure of the directional derivative and introducing a suitable subspace, we fully characterize the Bouligand subdifferential of the solution mapping. We then derive optimality conditions characterizing Bouligand-stationary and strongly-stationary points for discrete VI-constrained optimal control problems. A trust-region algorithm for solving these control problems is proposed based on the obtained characterizations, and a numerical experiment is presented to illustrate the main properties of both the solution and the proposed algorithm. |
| title | Bouligand Analysis and Discrete Optimal Control of Total Variation-Based Variational Inequalities |
| topic | Optimization and Control 49J40, 49K20, 49M29, 49J52, 90C33 |
| url | https://arxiv.org/abs/2504.09272 |