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Modified Combined Relaxation Method for General Monotone Equilibrium Problems in Hilbert Spaces

Author

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  • L. C. Zeng

    (Shanghai Normal University)

  • J. C. Yao

    (National Sun Yatsen University)

Abstract

In this paper, we study a class of general monotone equilibrium problems in a real Hilbert space which involves a monotone differentiable bifunction. For such a bifunction, a skew-symmetric type property with respect to the partial gradients is established. We suggest to solve this class of equilibrium problems with the modified combined relaxation method involving an auxiliary procedure. We prove the existence and uniqueness of the solution to the auxiliary variational inequality in the auxiliary procedure. Further, we prove also the weak convergence of the modified combined relaxation method by virtue of the monotonicity and the skew-symmetric type property.

Suggested Citation

  • L. C. Zeng & J. C. Yao, 2006. "Modified Combined Relaxation Method for General Monotone Equilibrium Problems in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 469-483, December.
    • Handle: RePEc:spr:joptap:v:131:y:2006:i:3:d:10.1007_s10957-006-9162-0
    DOI: 10.1007/s10957-006-9162-0
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    References listed on IDEAS

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    1. I. V. Konnov, 2001. "Combined Relaxation Method for Monotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 327-340, November.
    2. I. V. Konnov & S. Schaible & J. C. Yao, 2005. "Combined Relaxation Method for Mixed Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 126(2), pages 309-322, August.
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    Cited by:

    1. Phan Tu Vuong & Jean Jacques Strodiot, 2018. "The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces," Journal of Global Optimization, Springer, vol. 70(2), pages 477-495, February.
    2. M. Castellani & M. Giuli, 2010. "On Equivalent Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 157-168, October.
    3. Jean Strodiot & Phan Vuong & Thi Nguyen, 2016. "A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces," Journal of Global Optimization, Springer, vol. 64(1), pages 159-178, January.
    4. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
    5. 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.

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