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On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

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Abstract

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.

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Acknowledgements

We are grateful to Associate Editor Michael Patriksson and the referees for their insightful comments and suggestions that have helped us improve both the exposition and the content of this paper. The authors also thank Professor W. Takahashi of Tokyo Institute of Technology and Professor D. Kuroiwa of Shimane University for their helpful support.

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  1. Department of Electronics and Information Systems, Akita Prefectural University, 84-4 Yuri-Honjo, Akita, Japan

    Shin-ya Matsushita & Li Xu

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  1. Shin-ya Matsushita
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  2. Li Xu
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Correspondence to Shin-ya Matsushita.

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Communicated by Michael Patriksson.

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Matsushita, Sy., Xu, L. On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces. J Optim Theory Appl 161, 701–715 (2014). https://doi.org/10.1007/s10957-013-0460-z

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  • DOI: https://doi.org/10.1007/s10957-013-0460-z

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