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Simple proximal-type algorithms for equilibrium problems

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Abstract

This paper proposes two simple and elegant proximal-type algorithms to solve equilibrium problems with pseudo-monotone bifunctions in the setting of Hilbert spaces. The proposed algorithms use one proximal point evaluation of the bifunction at each iteration. Consequently, prove that the sequences of iterates generated by the first algorithm converge weakly to a solution of the equilibrium problem (assuming existence) and obtain a linear convergence rate under standard assumptions. We also design a viscosity version of the first algorithm and obtain its corresponding strong convergence result. Some popular existing algorithms in the literature are recovered. We finally give some numerical tests and compare our algorithms with some related ones to show the performance and efficiency of our proposed algorithms.

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Data sharing does not apply to this article as no datasets were generated or analyzed during the current study.

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The Matlab codes employed to run the numerical experiments are available on request.

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Acknowledgements

The authors are grateful to the handling editor and the two anonymous referees for their excellent suggestions and comments which have improved the earlier version of our manuscript greatly.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tiangong University, Tianjin, 300387, People’s Republic of China

    Yonghong Yao

  2. Center for Advanced Information Technology, Kyung Hee University, Seoul, 02447, South Korea

    Yonghong Yao

  3. Operational Research Center in Healthcare, Near East University, TRNC Mersin 10, 99138, Nicosia, Turkey

    Abubakar Adamu

  4. Mathematics Institute, African University of Science and Technology, Abuja, 900107, Nigeria

    Abubakar Adamu

  5. School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Yekini Shehu

  6. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Jen-Chih Yao

  7. Academy of Romanian Scientists, Bucharest, Romania

    Jen-Chih Yao

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  1. Yonghong Yao
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  2. Abubakar Adamu
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  3. Yekini Shehu
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  4. Jen-Chih Yao
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Correspondence to Yekini Shehu.

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Yao, Y., Adamu, A., Shehu, Y. et al. Simple proximal-type algorithms for equilibrium problems. J Glob Optim (2024). https://doi.org/10.1007/s10898-024-01377-1

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