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Convergence of a subgradient extragradient algorithm for solving monotone variational inequalities

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Abstract

In this paper, we introduce a new iterative algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a step size; an advantage of the algorithm is the computation of only one value of the mapping and one projection onto the admissible set per one iteration. The convergence of the algorithm is established without the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Also, we generalize the method with Bregman projection. Finally, some numerical experiments are presented to show the efficiency of the proposed algorithm.

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Acknowledgements

The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped to improve the original version of this paper.

Funding

This project is supported by the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2017JM1014).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710126, Shaanxi, China

    Jun Yang & Hongwei Liu

  2. School of Mathematics and Information Science, Xianyang Normal University, Xianyang, 712000, Shaanxi, China

    Jun Yang

  3. Xizang Minzu University, Xianyang, 712000, Shaanxi, China

    Guaiwei Li

Authors
  1. Jun Yang
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  2. Hongwei Liu
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  3. Guaiwei Li
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Correspondence to Jun Yang.

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Yang, J., Liu, H. & Li, G. Convergence of a subgradient extragradient algorithm for solving monotone variational inequalities. Numer Algor 84, 389–405 (2020). https://doi.org/10.1007/s11075-019-00759-x

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