IDEAS home Printed from https://ideas.repec.org/a/wly/jnljam/v2012y2012i1n924309.html

Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed‐Point Problems

Author

Listed:
  • Lu-Chuan Ceng
  • Ching-Feng Wen

Abstract

We investigate the problem of finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed‐point problem of a strictly pseudocontractive mapping in a real Hilbert space. Motivated by Nadezhkina and Takahashi′s hybrid‐extragradient method, we propose and analyze new hybrid‐extragradient iterative algorithm for finding a common solution. It is proven that three sequences generated by this algorithm converge strongly to the same common solution under very mild conditions. Based on this result, we also construct an iterative algorithm for finding a common fixed point of three mappings, such that one of these mappings is nonexpansive, and the other two mappings are strictly pseudocontractive mappings.

Suggested Citation

  • Lu-Chuan Ceng & Ching-Feng Wen, 2012. "Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed‐Point Problems," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:924309
    DOI: 10.1155/2012/924309
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2012/924309
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2012/924309?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
    ---><---

    References listed on IDEAS

    as
    1. Heinz H. Bauschke & Patrick L. Combettes, 2001. "A Weak-to-Strong Convergence Principle for Fejér-Monotone Methods in Hilbert Spaces," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 248-264, May.
    2. M. V. Solodov & B. F. Svaiter, 2000. "An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 214-230, May.
    3. N. Nadezhkina & W. Takahashi, 2006. "Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 128(1), pages 191-201, January.
    4. Eberhard Zeidler, 1985. "Nonlinear Functional Analysis and its Applications," Springer Books, Springer, number 978-1-4612-5020-3, October.
    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. 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.
    2. H. Zegeye & N. Shahzad, 2012. "Convergence Theorems for a Common Point of Solutions of Equilibrium and Fixed Point of Relatively Nonexpansive Multivalued Mapping Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    3. Yonghong Yao & Yeong-Cheng Liou & Ngai-Ching Wong, 2013. "Superimposed optimization methods for the mixed equilibrium problem and variational inclusion," Journal of Global Optimization, Springer, vol. 57(3), pages 935-950, November.
    4. Yanlai Song & Omar Bazighifan, 2022. "Regularization Method for the Variational Inequality Problem over the Set of Solutions to the Generalized Equilibrium Problem," Mathematics, MDPI, vol. 10(14), pages 1-20, July.
    5. S. Plubtieng & T. Thammathiwat, 2010. "A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities," Journal of Global Optimization, Springer, vol. 46(3), pages 447-464, March.
    6. Yanlai Song & Omar Bazighifan, 2022. "Two Regularization Methods for the Variational Inequality Problem over the Set of Solutions of the Generalized Mixed Equilibrium Problem," Mathematics, MDPI, vol. 10(16), pages 1-20, August.
    7. 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).
    8. Qian-Fen Gong & Dao-Jun Wen, 2014. "Further Investigation on the Relaxed Hybrid Steepest‐Descent Methods for Variational Inequalities with k‐Strict Pseudocontractions," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    9. Lu-Chuan Ceng & Juei-Ling Ho, 2014. "Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    10. Lei Yang & Kim-Chuan Toh, 2026. "Inexact Bregman Proximal Gradient Method and Its Inertial Variant with Absolute and Partial Relative Stopping Criteria," Mathematics of Operations Research, INFORMS, vol. 51(1), pages 686-714, January.
    11. A. E. Al-Mazrooei & A. Latif & J. C. Yao, 2014. "Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    12. Pourya Behmandpoor & Puya Latafat & Andreas Themelis & Marc Moonen & Panagiotis Patrinos, 2024. "SPIRAL: a superlinearly convergent incremental proximal algorithm for nonconvex finite sum minimization," Computational Optimization and Applications, Springer, vol. 88(1), pages 71-106, May.
    13. A. Moudafi, 2011. "Split Monotone Variational Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 275-283, August.
    14. Dan Butnariu & Elena Resmerita, 2006. "Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2006(1).
    15. Jonathan Eckstein & Paulo Silva, 2010. "Proximal methods for nonlinear programming: double regularization and inexact subproblems," Computational Optimization and Applications, Springer, vol. 46(2), pages 279-304, June.
    16. Dandan Yang & Chuanzhi Bai, 2014. "Three Weak Solutions for Nonlocal Fractional Laplacian Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    17. Anchalee Sripattanet & Atid Kangtunyakarn, 2019. "Convergence Theorem of Two Sequences for Solving the Modified Generalized System of Variational Inequalities and Numerical Analysis," Mathematics, MDPI, vol. 7(10), pages 1-18, October.
    18. M. Marques Alves & Dirk A. Lorenz & Emanuele Naldi, 2026. "A general framework for inexact splitting algorithms with relative errors and applications to Chambolle–Pock and Davis–Yin methods," Computational Optimization and Applications, Springer, vol. 93(2), pages 729-763, March.
    19. L. Zeng & J. Yao, 2009. "A hybrid extragradient method for general variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 141-158, March.
    20. Lu-Chuan Ceng & Cheng-Wen Liao & Chin-Tzong Pang & Ching-Feng Wen & Zhao-Rong Kong, 2014. "Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

    More about this item

    Statistics

    Access and download statistics

    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:wly:jnljam:v:2012:y:2012:i:1:n:924309. 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: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4185 .

    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.