Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization. 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 L1- and L∞-fitting terms as well as of state-constrained optimal control problems, where convergence can be guaranteed after introducing an (arbitrary small, still nonsmooth) Moreau--Yosida regularization. This is verified in numerical examples.

References in zbMATH (referenced in 10 articles , 1 standard article )

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  1. Afraites, Lekbir; Hadri, Aissam; Laghrib, Amine; Nachaoui, Mourad: A high order PDE-constrained optimization for the image denoising problem (2021)
  2. Annunziato, Mario; Borzì, Alfio: A Fokker-Planck approach to the reconstruction of a cell membrane potential (2021)
  3. Clason, Christian; Kunisch, Karl; Trautmann, Philip: Optimal control of the principal coefficient in a scalar wave equation (2021)
  4. Clason, Christian; Mazurenko, Stanislav; Valkonen, Tuomo: Primal-dual proximal splitting and generalized conjugation in non-smooth non-convex optimization (2021)
  5. Zeng, Shengda; Migórski, Stanisław; Khan, Akhtar A.: Nonlinear quasi-hemivariational inequalities: existence and optimal control (2021)
  6. Afraites, L.; Hadri, A.; Laghrib, A.: A denoising model adapted for impulse and Gaussian noises using a constrained-PDE (2020)
  7. Mazurenko, Stanislav; Jauhiainen, Jyrki; Valkonen, Tuomo: Primal-dual block-proximal splitting for a class of non-convex problems (2020)
  8. Migórski, Stanisław; Khan, Akhtar A.; Zeng, Shengda: Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems (2020)
  9. Clason, Christian; Mazurenko, Stanislav; Valkonen, Tuomo: Acceleration and global convergence of a first-order primal-dual method for nonconvex problems (2019)
  10. Clason, Christian; Valkonen, Tuomo: Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization (2017)