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A hybrid method without extrapolation step for solving variational inequality problems

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  • Yu. Malitsky
  • V. Semenov

Abstract

In this paper, we introduce a new method for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. The iterative process is based on well-known projection method and the hybrid (or outer approximation) method. However we do not use an extrapolation step in the projection method. The absence of one projection in our method is explained by slightly different choice of sets in the hybrid method. We prove a strong convergence of the sequences generated by our method. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Yu. Malitsky & V. Semenov, 2015. "A hybrid method without extrapolation step for solving variational inequality problems," Journal of Global Optimization, Springer, vol. 61(1), pages 193-202, January.
    • Handle: RePEc:spr:jglopt:v:61:y:2015:i:1:p:193-202
    DOI: 10.1007/s10898-014-0150-x
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    References listed on IDEAS

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    1. Alfredo Iusem & Mostafa Nasri, 2011. "Korpelevich’s method for variational inequality problems in Banach spaces," Journal of Global Optimization, Springer, vol. 50(1), pages 59-76, May.
    2. 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.
    3. Lu-Chuan Ceng & Nicolas Hadjisavvas & Ngai-Ching Wong, 2010. "Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems," Journal of Global Optimization, Springer, vol. 46(4), pages 635-646, April.
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    Cited by:

    1. Dang Hieu & Duong Viet Thong, 2018. "New extragradient-like algorithms for strongly pseudomonotone variational inequalities," Journal of Global Optimization, Springer, vol. 70(2), pages 385-399, February.
    2. Xiaomei Dong & Xingju Cai & Deren Han & Zhili Ge, 2020. "Solving a Class of Variational Inequality Problems with a New Inexact Strategy," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 37(01), pages 1-20, January.
    3. Yinglin Luo & Meijuan Shang & Bing Tan, 2020. "A General Inertial Viscosity Type Method for Nonexpansive Mappings and Its Applications in Signal Processing," Mathematics, MDPI, vol. 8(2), pages 1-18, February.
    4. Dang Hieu & Pham Ky Anh & Le Dung Muu, 2017. "Modified hybrid projection methods for finding common solutions to variational inequality problems," Computational Optimization and Applications, Springer, vol. 66(1), pages 75-96, January.
    5. Gang Cai & Qiao-Li Dong & Yu Peng, 2021. "Strong Convergence Theorems for Solving Variational Inequality Problems with Pseudo-monotone and Non-Lipschitz Operators," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 447-472, February.
    6. Pham Ngoc Anh, 2024. "Relaxed projection methods for solving variational inequality problems," Journal of Global Optimization, Springer, vol. 90(4), pages 909-930, December.
    7. 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.

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