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On the Strong Convergence of Halpern Type Proximal Point Algorithm

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Abstract

The main result of this paper is to prove the strong convergence of the sequence generated by the proximal point algorithm of Halpern type to a zero of a maximal monotone operator under the suitable assumptions on the parameters and error. The results extend some of the previous results or give some different conditions for convergence of the sequence. It is also indicated that when the maximal monotone operator is the subdifferential of a convex, proper, and lower semicontinuous function, the results extend all previous results in the literature. We also prove the boundedness of the sequence generated by the algorithm with a weak coercivity condition defined in the paper and without any additional assumptions on the parameters.

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Acknowledgements

This research was in part supported by a grant from University of Zanjan (No. 9041). The authors would like to thank the referee for valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran

    Hadi Khatibzadeh & Sajad Ranjbar

Authors
  1. Hadi Khatibzadeh
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  2. Sajad Ranjbar
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Correspondence to Hadi Khatibzadeh.

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Communicated by Nicolas Hadjisavvas.

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Khatibzadeh, H., Ranjbar, S. On the Strong Convergence of Halpern Type Proximal Point Algorithm. J Optim Theory Appl 158, 385–396 (2013). https://doi.org/10.1007/s10957-012-0213-4

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  • DOI: https://doi.org/10.1007/s10957-012-0213-4

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