Primal-Dual Extragradient Methods for Nonlinear Nonsmooth PDE-Constrained Optimization

Christian Clason, Tuomo Valkonen

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


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 (arbitrarily small, still nonsmooth) Moreau-Yosida regularization. This is verified in numerical examples.
Original languageEnglish (US)
Pages (from-to)1314-1339
Number of pages26
JournalSIAM Journal on Optimization
Issue number3
StatePublished - Jan 2017
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2021-04-06
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: While TV was in Cambridge, he was supported by the King Abdullah University of Science and Technology (KAUST) award KUK-I1-007-43, and EPSRC grants EP/J009539/1 “Sparse & Higher-order Image Restoration” and EP/M00483X/1 “Efficient Computational Tools for Inverse Imaging Problems.” Part of this work was also done while TV was in Quito, where he was supported by a Prometeo scholarship of the Senescyt (Ecuadorian Ministry of Science, Technology, Education, and Innovation). CC is supported by the German Science Foundation DFG under grant Cl 487/1-1.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software


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