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Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

Author

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  • Lu-Chuan Ceng

    (Department of Mathematics, Shanghai Normal University, Shanghai 200234, China)

  • Xiaolong Qin

    (General Education Center, National Yunlin University of Science and Technology, Douliou 64002, Taiwan)

  • Yekini Shehu

    (Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China)

  • Jen-Chih Yao

    (Research Center for Interneural Computing, China Medical University Hospital, Taichung 40447, Taiwan)

Abstract

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI).

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:881-:d:269630
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    References listed on IDEAS

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    1. Rapeepan Kraikaew & Satit Saejung, 2014. "Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 399-412, November.
    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 & Mihai Postolache & Ching-Feng Wen & Yonghong Yao, 2019. "Variational Inequalities Approaches to Minimization Problems with Constraints of Generalized Mixed Equilibria and Variational Inclusions," Mathematics, MDPI, vol. 7(3), pages 1-20, March.
    4. Xiaolong Qin & Nguyen Thai An, 2019. "Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets," Computational Optimization and Applications, Springer, vol. 74(3), pages 821-850, December.
    5. 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.
    6. Jiawei Chen & Elisabeth Köbis & Jen-Chih Yao, 2019. "Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 411-436, May.
    7. Lu-Chuan Ceng & Qamrul Ansari & Siegfried Schaible, 2012. "Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems," Journal of Global Optimization, Springer, vol. 53(1), pages 69-96, May.
    8. Zhao-Rong Kong & Lu-Chuan Ceng & Ching-Feng Wen, 2012. "Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-32, December.
    9. 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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