Quantitative results on a Halpern-type proximal point algorithm

Abstract

We apply proof mining methods to analyse a result of Boikanyo and Moroşanu on the strong convergence of a Halpern-type proximal point algorithm. As a consequence, we obtain quantitative versions of this result, providing uniform effective rates of asymptotic regularity and metastability.

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Acknowledgements

Laurenţiu Leuştean was partially supported by a grant of the Romanian Ministry of Research and Innovation, Program 1 - Development of the National RDI System, Subprogram 1.2 - Institutional Performance - Projects for Funding the Excellence in RDI, Contract Number 15PFE/2018. Pedro Pinto acknowledges and is thankful for the financial support of: FCT - Fundação para a Ciência e a Tecnologia under the Project UID/MAT/04561/2019; the research center Centro de Matemática, Aplicações Fundamentais com Investigação Operacional, Universidade de Lisboa; and the ‘Future Talents’ short-term scholarship at Technische Universität Darmstadt. The paper also benefited from discussions with Fernando Ferreira and Ulrich Kohlenbach.

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Correspondence to Laurenţiu Leuştean.

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Leuştean, L., Pinto, P. Quantitative results on a Halpern-type proximal point algorithm. Comput Optim Appl 79, 101–125 (2021). https://doi.org/10.1007/s10589-021-00263-w

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Keywords

  • Proximal point algorithm
  • Maximally monotone operators
  • Halpern iteration
  • Rates of convergence
  • Rates of metastability
  • Proof mining

Mathematics Subject Classification

  • 47H05
  • 47H09
  • 47J25
  • 03F10