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Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces

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Abstract

In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

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Acknowledgments

The authors would like to thank the Associate Editor and the anonymous referees for their really helpful and constructive comments that helped us very much in improving the paper.

Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.303.

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Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Pham Ngoc Anh

  2. Electric Power University, Hanoi, Vietnam

    T. V. Thang

  3. University of Transport Technology, Hanoi, Vietnam

    H. T. C. Thach

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  1. Pham Ngoc Anh
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  2. T. V. Thang
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  3. H. T. C. Thach
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Correspondence to Pham Ngoc Anh.

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Anh, P.N., Thang, T.V. & Thach, H.T.C. Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces. Numer Algor 87, 335–363 (2021). https://doi.org/10.1007/s11075-020-00968-9

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