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Error Sensitivity for Strongly Convergent Modifications of the Proximal Point Algorithm

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Abstract

The proximal point algorithm plays an important role in finding zeros of maximal monotone operators. It has however only weak convergence in the infinite-dimensional setting. In the present paper, we provide two contraction-proximal point algorithms. The strong convergence of the two algorithms is proved under two different accuracy criteria on the errors. A new technique of argument is used, and this makes sure that our conditions, which are sufficient for the strong convergence of the algorithms, are weaker than those used by several other authors.

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Acknowledgments

The authors were indebted to the anonymous referees for their helpful suggestions and comments on this manuscript. The authors were also grateful to Professor W. A. Kirk for his careful reading of the manuscript, which particularly improved the English language presentation of this manuscript. The research of Y. Wang was supported in part by National Natural Science Foundation of China with Grant No. 11271112. This work was also supported by the Scientific Research Foundation for Ph.D. of Henan Normal University (No. qd14144). The research of F.H. Wang was supported in part by the Program for Science and Technology Innovation Talents in the Universities of the Henan Province (No. 15HASTIT013).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

    Yamin Wang

  2. Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Sciences, Henan Normal University, Xinxiang, 453007, China

    Yamin Wang

  3. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

    Yamin Wang

  4. Department of Mathematics, Luoyang Normal University, Luoyang, 471022, China

    Fenghui Wang

  5. Department of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, China

    Hong-Kun Xu

  6. Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia

    Hong-Kun Xu

Authors
  1. Yamin Wang
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  2. Fenghui Wang
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  3. Hong-Kun Xu
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Correspondence to Hong-Kun Xu.

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Wang, Y., Wang, F. & Xu, HK. Error Sensitivity for Strongly Convergent Modifications of the Proximal Point Algorithm. J Optim Theory Appl 168, 901–916 (2016). https://doi.org/10.1007/s10957-015-0835-4

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  • DOI: https://doi.org/10.1007/s10957-015-0835-4

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