IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v154y2012i1d10.1007_s10957-011-9982-4.html
   My bibliography  Save this article

On Maingé’s Approach for Hierarchical Optimization Problems

Author

Listed:
  • Rapeepan Kraikaew

    (Khon Kaen University)

  • Satit Saejung

    (Khon Kaen University)

Abstract

This paper aims at investigating an iterative method for solving a system of variational inequalities with fixed-point set constraints. Our scheme can be regarded as a more general variant of the algorithm proposed by Maingé. Strong convergence results are established in the setting of Hilbert spaces. We propose an alternative analysis that allows us to relax some assumption imposed in his paper for convergence of the considered method. As a complementary result, we show how to adapt these processes to the case when the constraints involve operators belonging to the class of hemi-contractive mappings; this goes beyond the scope of Maingé’s result.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:1:d:10.1007_s10957-011-9982-4
    DOI: 10.1007/s10957-011-9982-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-011-9982-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-011-9982-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    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. Kassay, Gabor & Kolumban, Jozsef & Pales, Zsolt, 2002. "Factorization of Minty and Stampacchia variational inequality systems," European Journal of Operational Research, Elsevier, vol. 143(2), pages 377-389, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    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. Lu-Chuan Ceng & Adrian Petruşel & Jen-Chih Yao, 2019. "On Mann Viscosity Subgradient Extragradient Algorithms for Fixed Point Problems of Finitely Many Strict Pseudocontractions and Variational Inequalities," Mathematics, MDPI, vol. 7(10), pages 1-14, October.
    5. Yun-Ling Cui & Lu-Chuan Ceng & Fang-Fei Zhang & Cong-Shan Wang & Jian-Ye Li & Hui-Ying Hu & Long He, 2022. "Modified Mann-Type Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points Implicating Countably Many Nonexpansive Operators," Mathematics, MDPI, vol. 10(11), pages 1-26, June.
    6. 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.
    7. Tingting Cai & Dongmin Yu & Huanan Liu & Fengkai Gao, 2022. "Computational Analysis of Variational Inequalities Using Mean Extra-Gradient Approach," Mathematics, MDPI, vol. 10(13), pages 1-14, July.
    8. Lu-Chuan Ceng & Xiaoye Yang, 2019. "Some Mann-Type Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems Variational Inequalities and Fixed Point Problems," Mathematics, MDPI, vol. 7(3), pages 1-20, February.
    9. 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.
    10. Haiyun Zhou & Peiyuan Wang, 2014. "A Simpler Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 716-727, June.
    11. H.K. Xu, 2003. "An Iterative Approach to Quadratic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 116(3), pages 659-678, March.
    12. 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.
    13. 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.
    14. Jeong, Jae Ug, 2016. "Generalized viscosity approximation methods for mixed equilibrium problems and fixed point problems," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 168-180.
    15. Kanikar Muangchoo & Poom Kumam & Yeol Je Cho & Sompong Dhompongsa & Sakulbuth Ekvittayaniphon, 2019. "Approximating Fixed Points of Bregman Generalized α -Nonexpansive Mappings," Mathematics, MDPI, vol. 7(8), pages 1-28, August.
    16. L. C. Ceng & S. Schaible & J. C. Yao, 2008. "Implicit Iteration Scheme with Perturbed Mapping for Equilibrium Problems and Fixed Point Problems of Finitely Many Nonexpansive Mappings," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 403-418, November.
    17. Chainarong Khunpanuk & Bancha Panyanak & Nuttapol Pakkaranang, 2022. "A New Construction and Convergence Analysis of Non-Monotonic Iterative Methods for Solving ρ -Demicontractive Fixed Point Problems and Variational Inequalities Involving Pseudomonotone Mapping," Mathematics, MDPI, vol. 10(4), pages 1-29, February.
    18. 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.
    19. Lu-Chuan Ceng & Yekini Shehu & Jen-Chih Yao, 2022. "Modified Mann Subgradient-like Extragradient Rules for Variational Inequalities and Common Fixed Points Involving Asymptotically Nonexpansive Mappings," Mathematics, MDPI, vol. 10(5), pages 1-20, February.
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:154:y:2012:i:1:d:10.1007_s10957-011-9982-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.