Abstract
A semismooth Newton method (refered as DC–SSN) is proposed for the numerical solution of a class of nonconvex optimal control problems governed by linear elliptic partial differential equations. The nonconvex term in the cost functional arises from a Huber-type local regularization of the L q-quasinorm (q ∈ (0, 1)), therefore it promotes sparsity on the solution. The DC–SSN method solves the optimality system of the regularized problem resulting from the application of difference-of-convex functions programming tools.
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References
Kazufumi Ito and Karl Kunisch. Optimal control with L p( Ω), p ∈ [0, 1), control cost. SIAM Journal on Control and Optimization, 52(2):1251–1275, 2014.
Pedro Merino. A difference-of-convex functions approach for sparse pde optimal control problems with nonconvex costs. Computational Optimization and Applications, 74(1):225–258, 2019.
Georg Stadler. Elliptic optimal control problems with L 1-control cost and applications for the placement of control devices. Computational Optimization and Applications, 44(2):159, 2006.
Michael Ulbrich. Semismooth newton methods for operator equations in function spaces. SIAM Journal on Optimization, 13(3):805–841, 2002.
Acknowledgements
This research was funded by project PIMI-17-01 granted by Escuela Politécnica Nacional, Quito–Ecuador. Also, our research was carried out using the research computing facilities and services offered by Scientific Computing Laboratory of the Research Center on Mathematical Modeling: HPC–MODEMAT (http://www.hpcmodemat.epn.edu.ec), Escuela Politécnica Nacional, Quito–Ecuador.
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Merino, P. (2021). A Semismooth Newton Method for Regularized L q-quasinorm Sparse Optimal Control Problems. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_71
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DOI: https://doi.org/10.1007/978-3-030-55874-1_71
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