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New inertial forward-backward type for variational inequalities with Quasi-monotonicity

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Abstract

In this paper, we present a modification of the forward-backward splitting method with inertial extrapolation step and self-adaptive step sizes to solve variational inequalities in a quasi-monotone setting. Our proposed method involves one computation of the projection onto the feasible set and one evaluation of the operator per iteration, which is simpler than most methods available in the literature to solve similar problems. We first establish weak convergence result when the set of solutions of the Minty formulation of the variational inequality is nonempty in infinite dimensional Hilbert spaces under appropriate conditions. Next, we give linear convergence result when the operator is strongly pseudo-monotone. We also give numerical implementations of our proposed method and some comparisons with some other methods available in the literature.

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Acknowledgements

The authors are grateful to the associate editor and the anonymous referee for their insightful comments and suggestions which have improved greatly on the earlier version of the paper. This paper is dedicated to the loving memory of late Professor Charles Ejike Chidume (1947–2021).

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Authors and Affiliations

  1. Department of Mathematics, The Technion–Israel Institute of Technology, 3200003, Haifa, Israel

    Chinedu Izuchukwu

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Yekini Shehu

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

    Jen-Chih Yao

Authors
  1. Chinedu Izuchukwu
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  2. Yekini Shehu
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  3. Jen-Chih Yao
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Correspondence to Yekini Shehu.

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Izuchukwu, C., Shehu, Y. & Yao, JC. New inertial forward-backward type for variational inequalities with Quasi-monotonicity. J Glob Optim 84, 441–464 (2022). https://doi.org/10.1007/s10898-022-01152-0

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  • DOI: https://doi.org/10.1007/s10898-022-01152-0

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