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Auxiliary Principle Technique for Equilibrium Problems

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  • M. A. Noor

    (Etisalat College of Engineering)

Abstract

In this paper, we use the auxiliary principle technique to suggest and analyze a number of iterative methods for solving mixed quasiequilibrium problems. We prove that the convergence of these new methods requires either partially relaxed strongly monotonicity or peudomonotonicity, which is a weaker condition than monotonicity. Our proof of convergence is very simple as compared with others. These new results include several new and known results as special cases. Our results represent refinement and improvement of the previous known results for equilibrium and variational inequalities problems.

Suggested Citation

  • M. A. Noor, 2004. "Auxiliary Principle Technique for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 371-386, August.
    • Handle: RePEc:spr:joptap:v:122:y:2004:i:2:d:10.1023_b:jota.0000042526.24671.b2
    DOI: 10.1023/B:JOTA.0000042526.24671.b2
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    References listed on IDEAS

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    1. M.A. Noor, 2003. "Extragradient Methods for Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 475-488, June.
    2. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of Proximal Methods," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 311-326, May.
    3. M. A. Noor, 2003. "Iterative Methods for General Mixed Quasivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 123-136, October.
    4. M. A. Noor & K. I. Noor, 2004. "On General Mixed Quasivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 579-599, March.
    5. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 305-322, November.
    6. M.A. Noor, 2002. "Proximal Methods for Mixed Quasivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 453-459, November.
    7. M. A. Noor, 2003. "Resolvent Algorithms for Mixed Quasivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 137-149, October.
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    Cited by:

    1. Tran Quoc & Le Muu, 2012. "Iterative methods for solving monotone equilibrium problems via dual gap functions," Computational Optimization and Applications, Springer, vol. 51(2), pages 709-728, March.
    2. Maingé, Paul-Emile, 2010. "Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints," European Journal of Operational Research, Elsevier, vol. 205(3), pages 501-506, September.
    3. Nils Langenberg, 2012. "Interior point methods for equilibrium problems," Computational Optimization and Applications, Springer, vol. 53(2), pages 453-483, October.
    4. L. D. Muu & T. D. Quoc, 2009. "Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 185-204, July.
    5. N. N. Tam & J. C. Yao & N. D. Yen, 2008. "Solution Methods for Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 253-273, August.
    6. P. Anh & H. Le Thi, 2013. "An Armijo-type method for pseudomonotone equilibrium problems and its applications," Journal of Global Optimization, Springer, vol. 57(3), pages 803-820, November.
    7. P. Anh & T. Hai & P. Tuan, 2016. "On ergodic algorithms for equilibrium problems," Journal of Global Optimization, Springer, vol. 64(1), pages 179-195, January.
    8. M. A. Noor & K. I. Noor & E. Al-Said, 2011. "Auxiliary Principle Technique for Solving Bifunction Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 441-445, May.

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