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Parallel extragradient algorithms for multiple set split equilibrium problems in Hilbert spaces

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Abstract

In this paper, we introduce an extension of multiple set split variational inequality problem (Censor et al. Numer. Algor. 59, 301–323 2012) to multiple set split equilibrium problem (MSSEP) and propose two new parallel extragradient algorithms for solving MSSEP when the equilibrium bifunctions are Lipschitz-type continuous and pseudo-monotone with respect to their solution sets. By using extragradient method combining with cutting techniques, we obtain algorithms for these problems without using any product space. Under certain conditions on parameters, the iteration sequences generated by the proposed algorithms are proved to be weakly and strongly convergent to a solution of MSSEP. An application to multiple set split variational inequality problems and a numerical example and preliminary computational results are also provided.

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Acknowledgements

The authors would like to thank the referees very much for their constructive comments and suggestions which helped them very much in revising the paper. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016R1A2B4011589). The second author’s research was supported by a grand from Le Quy Don Technical University, Vietnam.

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Authors and Affiliations

  1. Department of Applied Mathematics, Pukyong National University, Busan, Korea

    Do Sang Kim

  2. Faculty of Information Technology, Le Quy Don Technical University, Hanoi, Vietnam

    Bui Van Dinh

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  1. Do Sang Kim
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  2. Bui Van Dinh
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Correspondence to Do Sang Kim.

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Kim, D.S., Van Dinh, B. Parallel extragradient algorithms for multiple set split equilibrium problems in Hilbert spaces. Numer Algor 77, 741–761 (2018). https://doi.org/10.1007/s11075-017-0338-5

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  • DOI: https://doi.org/10.1007/s11075-017-0338-5

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