Primal-Dual Extragradient Methods for Nonlinear Nonsmooth PDE-Constrained Optimization
Abstract
We study the extension of the Chambolle--Pock primal-dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces. Local convergence is shown under technical conditions including metric regularity of the corresponding primal-dual optimality conditions. We also show convergence for a Nesterov-type accelerated variant, provided one part of the functional is strongly convex. We show the applicability of the accelerated algorithm to examples of inverse problems with $L^1$ and $L^\infty$ fitting terms as well as of state-constrained optimal control problems, where convergence can be guaranteed after introducing an (arbitrarily small, still nonsmooth) Moreau--Yosida regularization. This is verified in numerical examples.
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© 2017, Society for Industrial and Applied Mathematics. This work is licensed under a Creative Commons Attribution 4.0 International License
History
Submitted: 21 June 2016
Accepted: 14 March 2017
Published online: 6 July 2017
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Secretaría de Educación Superior, Ciencia, Tecnología e Innovación
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Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : Cl 487/1-1
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King Abdullah University of Science and Technology https://doi.org/10.13039/501100004052 : KUK-I1-007-43
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Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/J009539/1, EP/M00483X/1
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