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Combined Relaxation Method for Mixed Equilibrium Problems

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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.

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References

  1. P.L. Lions B. Mercier (1979) ArticleTitleSplitting Algorithms for the Sum of Two Monotone Operators SIAM Journal of Numerical Analysis 16 964–979 Occurrence Handle10.1137/0716071

    Article  Google Scholar 

  2. J. Eckstein D. P. Bertsekas (1992) ArticleTitleOn the Douglas-Rachford Splitting Method and the Proximal-Point Algorithm for Maximal Monotone Operators Mathematical Programming 55 293–318 Occurrence Handle10.1007/BF01581204

    Article  Google Scholar 

  3. M. Patriksson (1999) Nonlinear Programming and Variational Inequality Problems: A Unified Approach Kluwer Academic Publishers Dordrecht Netherlands

    Google Scholar 

  4. P. D. Panagiotopoulos (1985) Inequality Problems in Mechanics and Their Applications Birkhäuser Massachusetts Boston

    Google Scholar 

  5. I. V. Konnov S. Schaible (2000) ArticleTitleDuality for Equilibrium Problems under Generalized Monotonicity Journal of Optimization Theory and Applications 104 395–408 Occurrence Handle10.1023/A:1004665830923

    Article  Google Scholar 

  6. I. V. Konnov (2001) Combined Relaxation Methods for Variational Inequalities Springer Verlag Berlin

    Google Scholar 

  7. I. V. Konnov (1993) ArticleTitleCombined Relaxation Methods for Finding Equilibrium Points and Solving Related Problems Russian Mathematics (Iz. VUZ) 37 44–51

    Google Scholar 

  8. N. Xiu J. Zhang (2003) ArticleTitleSome Recent Advances in Projection-Type Methods for Variational Inequalities Journal of Computational and Applied Mathematics 152 559–585 Occurrence Handle10.1016/S0377-0427(02)00730-6

    Article  Google Scholar 

  9. I. V. Konnov (1999) ArticleTitleA Combined Method for Variational Inequalities with Monotone Operators Computational Mathematics and Mathematical Physics 39 1051–1056

    Google Scholar 

  10. I. V. Konnov (2003) ArticleTitleApplication of the Proximal-Point Method to Nonmonotone Equilibrium Problems Journal of Optimization Theory and Applications 119 317–333 Occurrence Handle10.1023/B:JOTA.0000005448.12716.24

    Article  Google Scholar 

  11. C. Baiocchi A. Capelo (1984) Variational and Quasivariational Inequalities: Applications to Free Boundary Problems John Wiley and Sons New York, NY

    Google Scholar 

  12. B. T. Polyak (1987) Introduction to Optimization Nauka, Moscow, USSR, 1983 English Translation in Optimization Software New York, NY

    Google Scholar 

  13. M. C. Ferris (1991) ArticleTitleFinite Termination of the Proximal-Point Algorithm Mathematical Programming 50 359–366 Occurrence Handle10.1007/BF01594944

    Article  Google Scholar 

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Authors and Affiliations

  1. Professor, Department of Applied Mathematics, Kazan University, Kazan, Russia

    I. V. Konnov

  2. Professor, A.G. Anderson Graduate School of Management, University of California, Riverside, California

    S. Schaible

  3. Professor, Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    J. C. Yao

Authors
  1. I. V. Konnov
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  2. S. Schaible
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  3. J. C. Yao
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Additional information

S. Schaible - This author gratefully acknowledges partial support from the National Science Council of Taiwan.

J.C. Yao - His research was partially supported by Grant NSC 93-2115-M-110-011

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Konnov, I.V., Schaible, S. & Yao, J.C. Combined Relaxation Method for Mixed Equilibrium Problems. J Optim Theory Appl 126, 309–322 (2005). https://doi.org/10.1007/s10957-005-4716-0

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  • DOI: https://doi.org/10.1007/s10957-005-4716-0

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