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Combined Relaxation Method for Mixed Equilibrium Problems

Author

Listed:
  • I. V. Konnov

    (Kazan University)

  • S. Schaible

    (University of California)

  • J. C. Yao

    (National Sun Yat-Sen University)

Abstract

We consider a general class of equilibrium problems which involve a single-valued mapping and a nonsmooth bifunction. Such mixed equilibrium problems are solved with a combined relaxation method using an auxiliary iteration of a splitting-type method for constructing a separating hyperplane. We prove the convergence of the method under the assumption that the dual of the mixed equilibrium problem is solvable. Convergence rates are also derived.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:126:y:2005:i:2:d:10.1007_s10957-005-4716-0
    DOI: 10.1007/s10957-005-4716-0
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    Citations

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    Cited by:

    1. I. V. Konnov, 2014. "Right-Hand Side Decomposition for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 221-238, January.
    2. E. Allevi & A. Gnudi & I. V. Konnov & G. Oggioni, 2018. "Evaluating the effects of environmental regulations on a closed-loop supply chain network: a variational inequality approach," Annals of Operations Research, Springer, vol. 261(1), pages 1-43, February.
    3. I. V. Konnov, 1997. "A Class of Combined Iterative Methods for Solving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 677-693, September.
    4. Felipe Alvarez & Miguel Carrasco & Karine Pichard, 2005. "Convergence of a Hybrid Projection-Proximal Point Algorithm Coupled with Approximation Methods in Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 966-984, November.
    5. 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.
    6. M. Castellani & M. Giuli, 2010. "On Equivalent Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 157-168, October.
    7. 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.
    8. 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.
    9. Uthai Kamraksa & Rabian Wangkeeree, 2011. "Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces," Journal of Global Optimization, Springer, vol. 51(4), pages 689-714, December.
    10. N. J. Huang & J. Li & J. C. Yao, 2007. "Gap Functions and Existence of Solutions for a System of Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 201-212, May.
    11. Yao, Yonghong & Cho, Yeol Je & Liou, Yeong-Cheng, 2011. "Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems," European Journal of Operational Research, Elsevier, vol. 212(2), pages 242-250, July.
    12. 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.
    13. Alfredo Iusem & Mostafa Nasri, 2011. "Korpelevich’s method for variational inequality problems in Banach spaces," Journal of Global Optimization, Springer, vol. 50(1), pages 59-76, May.
    14. Minglu Ye & Yiran He, 2015. "A double projection method for solving variational inequalities without monotonicity," Computational Optimization and Applications, Springer, vol. 60(1), pages 141-150, January.
    15. I. V. Konnov, 1998. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 165-181, October.

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