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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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
Keywords
- Proximal point algorithm
- Contraction-proximal point algorithm
- Maximal monotone operator
- Resolvent
- Strong convergence