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Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities

Author

Listed:
  • L. C. Zeng

    (Shanghai Normal University)

  • N. C. Wong

    (National Sun Yat-Sen University)

  • J. C. Yao

    (National Sun Yat-Scn University)

Abstract

Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on H. We construct an iterative algorithm with variable parameters which generates a sequence {x n } from an arbitrary initial point x 0 ∊ H. The sequence {x n } is shown to converge in norm to the unique solution u ∗ of the variational inequality $$\langle F(u^{\ast}), v - u^{\ast}\rangle \geq 0, \quad \forall v \in C.$$

Suggested Citation

  • L. C. Zeng & N. C. Wong & J. C. Yao, 2007. "Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 51-69, January.
    • Handle: RePEc:spr:joptap:v:132:y:2007:i:1:d:10.1007_s10957-006-9068-x
    DOI: 10.1007/s10957-006-9068-x
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    References listed on IDEAS

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    1. 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.
    2. Jen-Chih Yao, 1994. "Variational Inequalities with Generalized Monotone Operators," Mathematics of Operations Research, INFORMS, vol. 19(3), pages 691-705, August.
    3. H.K. Xu, 2003. "An Iterative Approach to Quadratic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 116(3), pages 659-678, March.
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    Cited by:

    1. Nguyen Buong & Lam Thuy Duong, 2011. "An Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 513-524, December.
    2. Lu-Chuan Ceng & Jen-Chih Yao, 2013. "Existence theorems for generalized set-valued mixed (quasi-)variational inequalities in Banach spaces," Journal of Global Optimization, Springer, vol. 55(1), pages 27-51, January.
    3. Nguyen Buong & Nguyen Thi Hong Phuong, 2013. "Strong Convergence to Solutions for a Class of Variational Inequalities in Banach Spaces by Implicit Iteration Methods," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 399-411, November.
    4. Rapeepan Kraikaew & Satit Saejung, 2012. "On Maingé’s Approach for Hierarchical Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 71-87, July.
    5. Pham Ngoc Anh, 2023. "New Outer Proximal Methods for Solving Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 479-501, August.
    6. P. E. Maingé, 2008. "New Approach to Solving a System of Variational Inequalities and Hierarchical Problems," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 459-477, September.
    7. 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.
    8. L. C. Ceng & A. Petruşel, 2010. "Krasnoselski-Mann Iterations for Hierarchical Fixed Point Problems for a Finite Family of Nonself Mappings in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 617-639, September.

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