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

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  • Shin-ya Matsushita

    (Akita Prefectural University)

  • Li Xu

    (Akita Prefectural University)

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.

Suggested Citation

  • Shin-ya Matsushita & Li Xu, 2014. "On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 701-715, June.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:3:d:10.1007_s10957-013-0460-z
    DOI: 10.1007/s10957-013-0460-z
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    References listed on IDEAS

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    1. W. Takahashi & M. Toyoda, 2003. "Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 417-428, August.
    2. N.H. Xiu & J.Z. Zhang, 2002. "Local Convergence Analysis of Projection-Type Algorithms: Unified Approach," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 211-230, October.
    3. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
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    Cited by:

    1. Luong Nguyen & Qamrul Hasan Ansari & Xiaolong Qin, 2020. "Linear conditioning, weak sharpness and finite convergence for equilibrium problems," Journal of Global Optimization, Springer, vol. 77(2), pages 405-424, June.

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