Optimality conditions for a class of inverse optimal control problems with partial differential equations

Harder, Felix; Wachsmuth, Gerd

doi:10.1080/02331934.2018.1495205

Cited by 14 publications

(11 citation statements)

References 8 publications

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“…[5,26,27,35,36]. First steps regarding inverse optimal control of partial differential equations were carried out recently, see [16,20]. Therein, the authors heavily exploit the uniqueness of the lower level solution for any fixed upper level variable and the properties of the associated solution operator.…”

Section: Introductionmentioning

confidence: 99%

Solving inverse optimal control problems via value functions to global optimality

et al. 2019

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In this paper, we show how a special class of inverse optimal control problems of elliptic partial differential equations can be solved globally. Using the optimal value function of the underlying parametric optimal control problem, we transfer the overall hierarchical optimization problem into a nonconvex single-level one. Unfortunately, standard regularity conditions like Robinson's CQ are violated at all the feasible points of this surrogate problem. It is, however, shown that locally optimal solutions of the problem solve a Clarke-stationarity-type system. Moreover, we relax the feasible set of the surrogate problem iteratively by approximating the lower level optimal value function from above by piecewise affine functions. This allows us to compute globally optimal solutions of the original inverse optimal control problem. The global convergence of the resulting algorithm is shown theoretically and illustrated by means of a numerical example. KeywordsBilevel optimal control • Global optimization • Inverse optimal control • Optimality conditions • Solution algorithm This work is supported by the DFG Grant Analysis and Solution Methods for Bilevel Optimal Control Problems (Grant Nos. DE 650/10-1 and WA3636/4-1) within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).

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Section: Introductionmentioning

confidence: 99%

Solving inverse optimal control problems via value functions to global optimality

et al. 2019

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“…Second, when the lower level problem admits a unique solution for each fixed value of the upper level problem, one could use the associated solution operator to eliminate the lower level variable from the model. This approach has been used in [25,35,51] in order to derive necessary optimality conditions for bi-level optimal control problems. Finally, it is possible to exploit the optimal value function from the lower level problem in order to replace the bi-level problem equivalently by the so-called optimal value reformulation.…”

Section: Non-convex Non-smooth Bi-level Strategiesmentioning

confidence: 99%

A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification

Afraites

Hadri

Laghrib

et al. 2022

IPI

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<p style='text-indent:20px;'>We propose a new variational framework to remove a mixture of Gaussian and impulse noise from images. This framework is based on a non-convex PDE-constrained with a fractional-order operator. The non-convex norm is applied to the impulse component controlled by a weighted parameter <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, which depends on the level of the impulse noise and image feature. Furthermore, the fractional operator is used to preserve image texture and edges. In a first part, we study the theoretical properties of the proposed PDE-constrained, and we show some well-posdnees results. In a second part, after having demonstrated how to numerically find a minimizer, a proximal linearized algorithm combined with a Primal-Dual approach is introduced. Moreover, a bi-level optimization framework with a projected gradient algorithm is proposed in order to automatically select the parameter <inline-formula><tex-math id="M2">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>. Denoising tests confirm that the non-convex term and learned parameter <inline-formula><tex-math id="M3">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> lead in general to an improved reconstruction when compared to results of convex norm and other competitive denoising methods. Finally, we show extensive denoising experiments on various images and noise intensities and we report conventional numerical results which confirm the validity of the non-convex PDE-constrained, its analysis and also the proposed bi-level optimization with learning data.</p>

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“…Second, when the lower level problem admits a unique solution for each fixed value of the upper level problem, one could use the associated solution operator to eliminate the lower level variable from the model. This approach has been used in [14,17] in order to derive necessary optimality conditions for bi-level optimal control problems. Finally, it is possible to exploit the optimal value function from the lower level problem in order to replace the bi-level problem equivalently by the so-called optimal value reformulation.…”

Section: Basic Assumptionsmentioning

confidence: 99%

A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing

Nachaoui

Afraites

Hadri

et al. 2022

CPAA

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<p style='text-indent:20px;'>This paper introduce a novel optimization procedure to reduce mixture of Gaussian and impulse noise from images. This technique exploits a non-convex PDE-constrained characterized by a fractional-order operator. The used non-convex term facilitated the impulse component approximation controlled by a spatial parameter <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>. A non-convex and non-smooth bi-level optimization framework with a modified projected gradient algorithm is then proposed in order to learn the parameter <inline-formula><tex-math id="M2">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>. Denoising tests confirm that the non-convex term and learned parameter <inline-formula><tex-math id="M3">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> lead in general to an improved reconstruction when compared to results of convex norm and manual parameter <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> choice.</p>

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Optimality conditions for a class of inverse optimal control problems with partial differential equations

Cited by 14 publications

References 8 publications

Solving inverse optimal control problems via value functions to global optimality

Solving inverse optimal control problems via value functions to global optimality

A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification

A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing

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