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Regularization of proximal point algorithms in Hadamard manifolds

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Abstract

In this paper, we consider the regularization method for exact as well as for inexact proximal point algorithms for finding the singularities of maximal monotone set-valued vector fields. We prove that the sequences generated by these algorithms converge to an element of the set of singularities of a maximal monotone set-valued vector field. A numerical example is provided to illustrate the inexact proximal point algorithm with regularization. Applications of our results to minimization problems and saddle point problems are given in the setting of Hadamard manifolds.

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Acknowledgements

The authors are grateful to the reviewers for their valuable suggestions and corrections that improved the first draft of this paper. In this research, the first author was supported by a research Grant of DST-SERB No. EMR/2016/005124, and the last author was supported by a research Grant No. MOST 105-2115-M-039-002-MY3.

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Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Qamrul Hasan Ansari & Feeroz Babu

  2. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Qamrul Hasan Ansari

  3. Research Center for International Computing, China Medical University Hospital, China Medical University, Taichung, Taiwan

    Jen-Chih Yao

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  1. Qamrul Hasan Ansari
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  2. Feeroz Babu
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  3. Jen-Chih Yao
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Correspondence to Qamrul Hasan Ansari.

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Ansari, Q.H., Babu, F. & Yao, JC. Regularization of proximal point algorithms in Hadamard manifolds. J. Fixed Point Theory Appl. 21, 25 (2019). https://doi.org/10.1007/s11784-019-0658-2

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  • DOI: https://doi.org/10.1007/s11784-019-0658-2

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