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Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems

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  • L. C. Ceng

    (Shanghai Normal University
    Scientific Computing Key Laboratory of Shanghai Universities)

  • M. Teboulle

    (Tel Aviv University)

  • J. C. Yao

    (National Sun Yat-sen University)

Abstract

We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme.

Suggested Citation

  • L. C. Ceng & M. Teboulle & J. C. Yao, 2010. "Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 19-31, July.
    • Handle: RePEc:spr:joptap:v:146:y:2010:i:1:d:10.1007_s10957-010-9650-0
    DOI: 10.1007/s10957-010-9650-0
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    References listed on IDEAS

    as
    1. N. Nadezhkina & W. Takahashi, 2006. "Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 128(1), pages 191-201, January.
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    Cited by:

    1. Lateef Olakunle Jolaoso & Adeolu Taiwo & Timilehin Opeyemi Alakoya & Oluwatosin Temitope Mewomo, 2020. "A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 744-766, June.
    2. Lu-Chuan Ceng & Xiaolong Qin & Yekini Shehu & Jen-Chih Yao, 2019. "Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings," Mathematics, MDPI, vol. 7(10), pages 1-19, September.
    3. 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.
    4. 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.
    5. Yonghong Yao & Mihai Postolache, 2012. "Iterative Methods for Pseudomonotone Variational Inequalities and Fixed-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 273-287, October.
    6. Dang Van Hieu & Jean Jacques Strodiot & Le Dung Muu, 2020. "An Explicit Extragradient Algorithm for Solving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 476-503, May.

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