IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v190y2021i2d10.1007_s10957-021-01902-2.html
   My bibliography  Save this article

A Projected Extrapolated Gradient Method with Larger Step Size for Monotone Variational Inequalities

Author

Listed:
  • Xiaokai Chang

    (Lanzhou University of Technology)

  • Jianchao Bai

    (Northwestern Polytechnical University)

Abstract

A projected extrapolated gradient method is designed for solving monotone variational inequality in Hilbert space. Requiring local Lipschitz continuity of the operator, our proposed method improves the value of the extrapolated parameter and admits larger step sizes, which are predicted based a local information of the involved operator and corrected by bounding the distance between each pair of successive iterates. The correction will be implemented when the distance is larger than a given constant and its main cost is to compute a projection onto the feasible set. In particular, when the operator is the gradient of a convex function, the correction step is not necessary. We establish the convergence and ergodic convergence rate in theory under the larger range of parameters. Related numerical experiments illustrate the improvements in efficiency from the larger step sizes.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:190:y:2021:i:2:d:10.1007_s10957-021-01902-2
    DOI: 10.1007/s10957-021-01902-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-021-01902-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-021-01902-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Duong Viet Thong & Vu Tien Dung & Hoang Thi Thanh Tam, 2025. "A Self Adaptive Projected Gradient Method for Solving Non-Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-27, July.
    2. 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.
    3. Duong Thong & Pham Anh & Vu Dung, 2025. "Relaxed Two-Step Inertial Tseng’s Extragradient Method for Nonmonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 205(1), pages 1-27, April.
    4. Duong Viet Thong, 2025. "Some New Conditions for Solving Variational Inequalities Without Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-20, August.
    5. Shin-ya Matsushita & Li Xu, 2014. "On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 701-715, June.
    6. Jamilu Abubakar & Poom Kumam & Habib ur Rehman & Abdulkarim Hassan Ibrahim, 2020. "Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator," Mathematics, MDPI, vol. 8(4), pages 1-25, April.
    7. Yanlai Song & Omar Bazighifan, 2022. "A New Alternative Regularization Method for Solving Generalized Equilibrium Problems," Mathematics, MDPI, vol. 10(8), pages 1-14, April.
    8. Yanlai Song & Omar Bazighifan, 2022. "Modified Inertial Subgradient Extragradient Method with Regularization for Variational Inequality and Null Point Problems," Mathematics, MDPI, vol. 10(14), pages 1-17, July.
    9. Dang Hieu, 2017. "New subgradient extragradient methods for common solutions to equilibrium problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 571-594, July.
    10. Chinedu Izuchukwu & Yekini Shehu, 2021. "New Inertial Projection Methods for Solving Multivalued Variational Inequality Problems Beyond Monotonicity," Networks and Spatial Economics, Springer, vol. 21(2), pages 291-323, June.
    11. Timilehin O. Alakoya & Oluwatosin T. Mewomo & Yekini Shehu, 2022. "Strong convergence results for quasimonotone variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(2), pages 249-279, April.
    12. 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.
    13. 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.
    14. Yekini Shehu & Olaniyi S. Iyiola & Duong Viet Thong & Nguyen Thi Cam Van, 2021. "An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(2), pages 213-242, April.
    15. Dang Hieu & Pham Ky Anh & Le Dung Muu, 2019. "Modified extragradient-like algorithms with new stepsizes for variational inequalities," Computational Optimization and Applications, Springer, vol. 73(3), pages 913-932, July.
    16. Habib ur Rehman & Kanokwan Sitthithakerngkiet & Thidaporn Seangwattana, 2024. "A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities," Mathematics, MDPI, vol. 13(1), pages 1-32, December.
    17. Shanshan Xu & Songxiao Li, 2025. "A Strongly Convergent Alternated Inertial Algorithm for Solving Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-35, August.
    18. P. E. Maingé & M. L. Gobinddass, 2016. "Convergence of One-Step Projected Gradient Methods for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 146-168, October.
    19. Gang Cai & Aviv Gibali & Olaniyi S. Iyiola & Yekini Shehu, 2018. "A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 219-239, July.
    20. Boţ, R.I. & Csetnek, E.R. & Vuong, P.T., 2020. "The forward–backward–forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces," European Journal of Operational Research, Elsevier, vol. 287(1), pages 49-60.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:190:y:2021:i:2:d:10.1007_s10957-021-01902-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.