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Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

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  • Songnian He
  • Caiping Yang

Abstract

Consider the variational inequality of finding a point satisfying the property , for all , where is the intersection of finite level sets of convex functions defined on a real Hilbert space and is an -Lipschitzian and -strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of . Since our algorithm avoids calculating the projection (calculating by computing several sequences of projections onto half-spaces containing the original domain ) directly and has no need to know any information of the constants and , the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.

Suggested Citation

  • Songnian He & Caiping Yang, 2013. "Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, May.
    • Handle: RePEc:hin:jnlaaa:942315
    DOI: 10.1155/2013/942315
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    Cited by:

    1. Cholamjiak, Watcharaporn & Dutta, Hemen, 2022. "Viscosity modification with parallel inertial two steps forward-backward splitting methods for inclusion problems applied to signal recovery," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    2. Suthep Suantai & Suparat Kesornprom & Prasit Cholamjiak, 2019. "Modified Proximal Algorithms for Finding Solutions of the Split Variational Inclusions," Mathematics, MDPI, vol. 7(8), pages 1-17, August.
    3. Peichao Duan & Xubang Zheng & Jing Zhao, 2018. "Strong Convergence Theorems of Viscosity Iterative Algorithms for Split Common Fixed Point Problems," Mathematics, MDPI, vol. 7(1), pages 1-14, December.
    4. Xinglong Wang & Jing Zhao & Dingfang Hou, 2019. "Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem," Mathematics, MDPI, vol. 7(2), pages 1-17, January.
    5. Yuanheng Wang & Tiantian Xu & Jen-Chih Yao & Bingnan Jiang, 2022. "Self-Adaptive Method and Inertial Modification for Solving the Split Feasibility Problem and Fixed-Point Problem of Quasi-Nonexpansive Mapping," Mathematics, MDPI, vol. 10(9), pages 1-15, May.
    6. Kamonrat Sombut & Kanokwan Sitthithakerngkiet & Areerat Arunchai & Thidaporn Seangwattana, 2023. "An Inertial Forward–Backward Splitting Method for Solving Modified Variational Inclusion Problems and Its Application," Mathematics, MDPI, vol. 11(9), pages 1-16, April.
    7. Suthep Suantai & Nontawat Eiamniran & Nattawut Pholasa & Prasit Cholamjiak, 2019. "Three-Step Projective Methods for Solving the Split Feasibility Problems," Mathematics, MDPI, vol. 7(8), pages 1-15, August.
    8. Adamu, A. & Kitkuan, D. & Padcharoen, A. & Chidume, C.E. & Kumam, P., 2022. "Inertial viscosity-type iterative method for solving inclusion problems with applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 445-459.
    9. Chanjuan Pan & Yuanheng Wang, 2019. "Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces," Mathematics, MDPI, vol. 7(2), pages 1-12, February.
    10. Songnian He & Qiao-Li Dong, 2018. "The Combination Projection Method for Solving Convex Feasibility Problems," Mathematics, MDPI, vol. 6(11), pages 1-13, November.
    11. Suthep Suantai & Kunrada Kankam & Damrongsak Yambangwai & Watcharaporn Cholamjiak, 2022. "A Modified Inertial Parallel Viscosity-Type Algorithm for a Finite Family of Nonexpansive Mappings and Its Applications," Mathematics, MDPI, vol. 10(23), pages 1-21, November.

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