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Strong Convergence of Regularized New Proximal Point Algorithms

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Abstract

We consider the regularization of two proximal point algorithms (PPA) with errors for a maximal monotone operator in a real Hilbert space, previously studied, respectively, by Xu, and by Boikanyo and Morosanu, where they assumed the zero set of the operator to be nonempty. We provide a counterexample showing an error in Xu’s theorem, and then we prove its correct extended version by giving a necessary and sufficient condition for the zero set of the operator to be nonempty and showing the strong convergence of the regularized scheme to a zero of the operator. This will give a first affirmative answer to the open question raised by Boikanyo and Morosanu concerning the design of a PPA, where the error sequence tends to zero and a parameter sequence remains bounded. Then, we investigate the second PPA with various new conditions on the parameter sequences and prove similar theorems as above, providing also a second affirmative answer to the open question of Boikanyo and Morosanu. Finally, we present some applications of our new convergence results to optimization and variational inequalities.

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Acknowledgements

The authors are grateful to the editor and the referees for valuable suggestions leading to the improvement of the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX, 79968, USA

    Behzad Djafari Rouhani

  2. Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran

    Sirous Moradi

Authors
  1. Behzad Djafari Rouhani
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  2. Sirous Moradi
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Correspondence to Behzad Djafari Rouhani.

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Communicated by Qamrul Hasan Ansari.

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Djafari Rouhani, B., Moradi, S. Strong Convergence of Regularized New Proximal Point Algorithms. J Optim Theory Appl 181, 864–882 (2019). https://doi.org/10.1007/s10957-019-01497-9

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