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Relaxed Single Projection Methods for Solving Bilevel Variational Inequality Problems in Hilbert Spaces

Author

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  • Ferdinard U. Ogbuisi

    (University of Nigeria)

  • Yekini Shehu

    (Zhejiang Normal University)

  • Jen-Chih Yao

    (China Medical University Hospital, China Medical University
    National Sun Yat-sen University)

Abstract

In this paper, we first propose a relaxed regularization projection method involving only a single projection for solving monotone bilevel variational inequality problem in Hilbert spaces and secondly we give an alternated inertial version of the first algorithm. The two proposed algorithms involve self adaptive step-sizes and the algorithms can easily be implemented without the prior knowledge of Lipschitz and strongly monotone constants of operators. Under some mild standard assumptions, we obtain the strong convergence of the two algorithms to the unique solution of the bilevel equilibrium problem. Moreover, some interesting numerical experiments are given to demonstrate the applicability of the results and also to compare with existing algorithms.

Suggested Citation

  • Ferdinard U. Ogbuisi & Yekini Shehu & Jen-Chih Yao, 2023. "Relaxed Single Projection Methods for Solving Bilevel Variational Inequality Problems in Hilbert Spaces," Networks and Spatial Economics, Springer, vol. 23(3), pages 641-678, September.
    • Handle: RePEc:kap:netspa:v:23:y:2023:i:3:d:10.1007_s11067-023-09594-z
    DOI: 10.1007/s11067-023-09594-z
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    References listed on IDEAS

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    1. 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.
    2. Abdellatif Moudafi, 2010. "Proximal methods for a class of bilevel monotone equilibrium problems," Journal of Global Optimization, Springer, vol. 47(2), pages 287-292, June.
    3. Franck Iutzeler & Jérôme Malick, 2018. "On the Proximal Gradient Algorithm with Alternated Inertia," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 688-710, March.
    4. J. Glackin & J. G. Ecker & M. Kupferschmid, 2009. "Solving Bilevel Linear Programs Using Multiple Objective Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 197-212, February.
    5. 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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