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A Modified Projected Gradient Method for Monotone Variational Inequalities

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  • Jun Yang

    (Xidian University
    Xianyang Normal University)

  • Hongwei Liu

    (Xidian University)

Abstract

In this paper, we investigate and analyze classical variational inequalities with Lipschitz continuous and monotone mapping in real Hilbert space. The projected reflected gradient method, with varying step size, requires at most two projections onto the feasible set and one value of the mapping per iteration. We modify the method with a simple structure; a weak convergence theorem for our algorithm is proved without any requirement of additional projections and the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Preliminary results from numerical experiments are performed.

Suggested Citation

  • Jun Yang & Hongwei Liu, 2018. "A Modified Projected Gradient Method for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 197-211, October.
    • Handle: RePEc:spr:joptap:v:179:y:2018:i:1:d:10.1007_s10957-018-1351-0
    DOI: 10.1007/s10957-018-1351-0
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    References listed on IDEAS

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    1. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
    2. Javad Balooee, 2013. "Projection Method Approach for General Regularized Non-convex Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 192-209, October.
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    Cited by:

    1. Duong Viet Thong & Xiao-Huan Li & Vu Tien Dung & Pham Thi Huong Huyen & Hoang Thi Thanh Tam, 2024. "Using Double Inertial Steps Into the Single Projection Method with Non-monotonic Step Sizes for Solving Pseudomontone Variational Inequalities," Networks and Spatial Economics, Springer, vol. 24(1), pages 1-26, March.
    2. Jun Yang & Hongwei Liu, 2020. "A self-adaptive method for pseudomonotone equilibrium problems and variational inequalities," Computational Optimization and Applications, Springer, vol. 75(2), pages 423-440, March.
    3. Yonghong Yao & Mihai Postolache & Jen-Chih Yao, 2019. "Iterative Algorithms for Pseudomonotone Variational Inequalities and Fixed Point Problems of Pseudocontractive Operators," Mathematics, MDPI, vol. 7(12), pages 1-13, December.
    4. Hongwei Liu & Jun Yang, 2020. "Weak convergence of iterative methods for solving quasimonotone variational inequalities," Computational Optimization and Applications, Springer, vol. 77(2), pages 491-508, November.
    5. Duong Viet Thong & Phan Tu Vuong & Pham Ky Anh & Le Dung Muu, 2022. "A New Projection-type Method with Nondecreasing Adaptive Step-sizes for Pseudo-monotone Variational Inequalities," Networks and Spatial Economics, Springer, vol. 22(4), pages 803-829, December.
    6. Xiaokai Chang & Jianchao Bai, 2021. "A Projected Extrapolated Gradient Method with Larger Step Size for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 602-627, August.
    7. Yonghong Yao & Naseer Shahzad & Jen-Chih Yao, 2020. "Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators," Mathematics, MDPI, vol. 8(4), pages 1-15, March.

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