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Iterative Methods for Triple Hierarchical Variational Inequalities in Hilbert Spaces

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  • Lu-Chuan Ceng

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

  • Qamrul Hasan Ansari

    (Aligarh Muslim University)

  • Jen-Chih Yao

    (Kaohsiung Medical University)

Abstract

In this paper, we consider a variational inequality with a variational inequality constraint over a set of fixed points of a nonexpansive mapping called triple hierarchical variational inequality. We propose two iterative methods, one is implicit and another one is explicit, to compute the approximate solutions of our problem. We present an example of our problem. The convergence analysis of the sequences generated by the proposed methods is also studied.

Suggested Citation

  • Lu-Chuan Ceng & Qamrul Hasan Ansari & Jen-Chih Yao, 2011. "Iterative Methods for Triple Hierarchical Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 489-512, December.
    • Handle: RePEc:spr:joptap:v:151:y:2011:i:3:d:10.1007_s10957-011-9882-7
    DOI: 10.1007/s10957-011-9882-7
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    References listed on IDEAS

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    1. Patrick Jaillet & Damien Lamberton & Bernard Lapeyre, 1990. "Variational inequalities and the pricing of American options," Post-Print hal-01667008, HAL.
    2. H. K. Xu & T. H. Kim, 2003. "Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 185-201, October.
    3. Giuseppe Marino & Hong-Kun Xu, 2011. "Explicit Hierarchical Fixed Point Approach to Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(1), pages 61-78, April.
    4. H. Iiduka, 2009. "Hybrid Conjugate Gradient Method for a Convex Optimization Problem over the Fixed-Point Set of a Nonexpansive Mapping," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 463-475, March.
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