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On the convergence of algorithms with Tikhonov regularization terms

Abstract

We consider the strongly convergent modified versions of the Krasnosel’skiĭ-Mann, the forward-backward and the Douglas-Rachford algorithms with Tikhonov regularization terms, introduced by Radu Boţ, Ernö Csetnek and Dennis Meier. We obtain quantitative information for these modified iterations, namely rates of asymptotic regularity and metastability. Furthermore, our arguments avoid the use of sequential weak compactness and use only a weak form of the projection argument.

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Notes

  1. 1.

    The standard definition asks for \(\alpha \in (0,1)\). With this extension, 1-averaged is just another way of saying nonexpansive.

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Acknowledgements

Both authors acknowledge the support of FCT - Fundação para a Ciência e Tecnologia under the projects: UIDB/04561/2020 and UIDP/04561/2020, and the research center Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Universidade de Lisboa. The second author also acknowledges the support of the ‘Future Talents’ short-term scholarship at Technische Universität Darmstadt. The authors also like to thank the suggestions made by the anonymous referees.

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Correspondence to Bruno Dinis.

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Appendix

Appendix

Fig. 1
figure1

Constants used

Fig. 2
figure2

Functions used

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Dinis, B., Pinto, P. On the convergence of algorithms with Tikhonov regularization terms. Optim Lett 15, 1263–1276 (2021). https://doi.org/10.1007/s11590-020-01635-7

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Keywords

  • Fixed points of nonexpansive mappings
  • Tikhonov regularization
  • Splitting algorithms
  • Metastability
  • Asymptotic regularity