Computational Errors of the Extragradient Method for Equilibrium Problems

  • Pham Ngoc Anh
  • Nguyen Duc Hien
  • Pham Minh Tuan
Article
  • 24 Downloads

Abstract

Our aim in this paper is to study variants and computational errors of the extragradient method for solving equilibrium problems. First, we consider convergence of the method when domains in the auxiliary subproblems of the extragradient algorithm are replaced by outer and inner approximation polyhedra. Then, computational errors are showed under the asymptotic optimality condition, but the bifunction must satisfy certain Lipschitz-type continuous conditions. Next, by using Armijo-type linesearch techniques commonly used in variational inequalities, we obtain an approximation linesearch algorithm without Lipschitz continuity. Convergence analysis of the algorithms is considered under mild conditions on the iterative parameters.

Keywords

Equilibrium problems Semicontinuous Extragradient algorithm Computational errors 

Mathematics Subject Classification

65 K10 90 C25 47 H05 47 H09 

Notes

Acknowledgements

We are very grateful to the editor and anonymous referees for their comments that helped us very much in improving the paper. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2017.15.

References

  1. 1.
    Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62, 271–283 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anh, P.N.: A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems. Acta Math. Vietnam. 34, 183–200 (2009)MathSciNetMATHGoogle Scholar
  3. 3.
    Anh, P.N.: An LQP regularization method for equilibrium problems on polyhedral. Vietnam J. Math. 36, 209–228 (2008)MathSciNetGoogle Scholar
  4. 4.
    Anh, P.N., Hien, N.D.: Fixed point solution methods for solving equilibrium problems. Bull. Korean Math. Soc. 51, 479–499 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Anh, P.N., Kim, J.K.: Outer approximation algorithms for pseudomonotone equilibrium problems. Comput. Math. Appl. 61, 2588–2595 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Anh, P.N., Kim, J.K.: An interior proximal cutting hyperplane method for equilibrium problems. J. Inequal. Appl. (2012).  https://doi.org/10.1186/1029-242X-2012-99
  7. 7.
    Anh, P.N., Kim, J.K., Hien, N.D.: A cutting hyperplane method for solving pseudomonotone non-lipschitzian equilibrium problems. J. Inequal. Appl. (2012).  https://doi.org/10.1186/1029-242X-2012-288
  8. 8.
    Anh, P.N., Kuno, T.: A cutting hyperplane method for generalized monotone nonlipschitzian multivalued variational inequalities. In: Bock, H.G., Phu, H.X., Rannacher, R., Schloder, J.P. (eds.) Modeling, Simulation and Optimization of Complex Processes. Springer, Berlin (2012)Google Scholar
  9. 9.
    Anh, P.N., Le Thi, H.A.: Outer-Interior Proximal Projection Methods for Multivalued Variational Inequalities. ACTA Math. Vietnam.  https://doi.org/10.1007/s40306-015-0165-5
  10. 10.
    Anh, P.N., Le Thi, H.A.: An armijo-type method for pseudomonotone equilibrium problems and its applications. J. Glob. Optim. 57, 803–820 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Antipin, A.S.: The convergence of proximal methods to fixed points of extremal mappings and estimates of their rates of convergence. Comput. Math. Math. Phys. 35, 539–551 (1995)MathSciNetMATHGoogle Scholar
  12. 12.
    Attouch, H.: Variational Convergence for Functions and Operators. Pitman, London (1984)MATHGoogle Scholar
  13. 13.
    Blum, E., Oettli, W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994)MATHGoogle Scholar
  14. 14.
    Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61(9), 1119–1132 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cohen, G.: Auxiliary problem principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dafermos, S.: Exchange price equilibria and variational inequalities. Math. Progamm. 46, 391–402 (1990)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Daniele, P., Giannessi, F., Maugeri, A.: Equilibrium Problems and Variational Models. Kluwer, Dordrecht (2003)CrossRefMATHGoogle Scholar
  18. 18.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, NewYork (2003)MATHGoogle Scholar
  19. 19.
    Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58, 251–261 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Iiduka, H., Yamada, I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model Algorithms 8, 1–18 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52, 301–316 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)MATHGoogle Scholar
  23. 23.
    Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F., Maugeri, A. (eds.) Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (2003)Google Scholar
  24. 24.
    Moudafi, A.: Proximal point algorithm extended to equilibrium problem. J. Nat. Geom. 15, 91–100 (1999)MathSciNetMATHGoogle Scholar
  25. 25.
    Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Glob. Optim. (2012).  https://doi.org/10.1007/s10898-011-9814-y
  27. 27.
    Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms to equilibrium Problems. J. Glob. Optim. 52, 139–159 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Taji, K., Fukushima, M.: A new merit function and a successive quadratic programming algorithm for variational inequality problem. SIAM J. Optim. 6, 704–713 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  • Pham Ngoc Anh
    • 1
  • Nguyen Duc Hien
    • 2
  • Pham Minh Tuan
    • 3
  1. 1.Department of Scientific FundamentalsPosts and Telecommunications Institute of TechnologyHanoiVietnam
  2. 2.Office of Scientific Research and TechnologyDuy Tan UniversityDa NangVietnam
  3. 3.Academy of Military Science and TechnologyHanoiVietnam

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