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A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications

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Abstract

In this paper, we propose a new algorithm for solving a bilevel equilibrium problem in a real Hilbert space. In contrast to most other projection-type algorithms, which require to solve subproblems at each iteration, the subgradient method proposed in this paper requires only to calculate, at each iteration, two subgradients of convex functions and one projection onto a convex set. Hence, our algorithm has a low computational cost. We prove a strong convergence theorem for the proposed algorithm and apply it for solving the equilibrium problem over the fixed point set of a nonexpansive mapping. Some numerical experiments and comparisons are given to illustrate our results. Also, an application to Nash–Cournot equilibrium models of a semioligopolistic market is presented.

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References

  1. Fan, K.: A Minimax Inequality and Applications, Inequalities III, pp. 103–113. Academic Press, New York (1972)

    Google Scholar 

  2. Iusem, A.N.: On some properties of paramonotone operators. J. Convex Anal. 5, 269–278 (1998)

    MATH  MathSciNet  Google Scholar 

  3. Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)

    MATH  MathSciNet  Google Scholar 

  4. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MATH  MathSciNet  Google Scholar 

  5. Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159–1166 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Korpolevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Matemat. Metody 12, 747–756 (1976)

    MathSciNet  Google Scholar 

  7. Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    Article  MathSciNet  Google Scholar 

  8. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 34, 1814–1830 (1996)

    Article  MathSciNet  Google Scholar 

  9. Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces. J. Glob. Optim. 59, 173–190 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Anh, P.K., Hieu, D.V.: Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems. Vietnam J. Math. 44, 351–374 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  11. Anh, P.N., Hai, T.N., Tuan, P.M.: On ergodic algorithms for equilibrium problems. J. Glob. Optim. 64, 179–195 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hai, T.N., Vinh, N.T.: Two new splitting algorithms for equilibrium problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 111, 1051–1069 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  14. 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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Colao, V., Marino, G., Xu, H.K.: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  17. Saeidi, S.: Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings. Nonlinear Anal. 70, 4195–4208 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Anh, P.N., Muu, L.D.: A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems. Optim. Lett. 8, 727–738 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  19. Iiduka, H.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227–242 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Cegielski, A., Zalas, R.: Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators. Numer. Funct. Anal. Optim. 34, 255–283 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  22. Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Dempe, S., Dutta, J., Lohse, S.: Optimality conditions for bilevel programming problems. Optimization 55, 505–524 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Dinh, B.V., Hung, P.G., Muu, L.D.: Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems. Numer. Funct. Anal. Optim. 35, 539–563 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  25. Iiduka, H.: A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping. Optimization 59, 873–885 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Iiduka, H., Yamada, I.: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881–1893 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Muu, L.D., Oettli, W.: Optimization over equilibrium sets. Optimization 49, 179–189 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  28. Takahashi, A., Yamada, I.: Parallel algorithms for variational inequalities over the Cartesian product of the intersections of the fixed point sets of nonexpansive mappings. J. Approx. Theory 153, 139–160 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Stud. Comput. Math. 8, 473–504 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Duc, P.M., Muu, L.D.: A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization 65, 1855–1866 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  31. Quy, N.V.: An algorithm for a bilevel problem with equilibrium and fixed point constraints. Optimization 64, 1–17 (2014)

    MathSciNet  Google Scholar 

  32. Maingé, P.E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur. J. Oper. Res. 205, 501–506 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  33. Yen, L.H., Muu, L.D., Huyen, N.T.T.: An algorithm for a class of split feasibility problems: application to a model in electricity production. Math. Methods Oper. Res. 84, 549–565 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  34. Iusem, A.N.: On the maximal monotonicity of diagonal subdifferential operators. J. Convex Anal. 18, 489–503 (2011)

    MATH  MathSciNet  Google Scholar 

  35. Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  36. Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model. J. Optim. Theory Appl. 142, 185–204 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  37. Van, N.T.T., Strodiot, J.J., Nguyen, V.H.: The interior proximal extragradient method for solving equilibrium problems. J. Glob. Optim. 44, 175–192 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  38. Combettes, P.L.: Quasi-Fejérian analysis of some optimization algorithms. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms for Feasibility and Optimization, pp. 115–152. Elsevier, New York (2001)

    Chapter  Google Scholar 

  39. Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal. 16, 899–912 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  40. Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  41. Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 52, 139–159 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank two anonymous referees and the editor for their constructive comments which helped to improve the paper.

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Authors and Affiliations

  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam

    Le Quang Thuy & Trinh Ngoc Hai

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  1. Le Quang Thuy
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  2. Trinh Ngoc Hai
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Correspondence to Trinh Ngoc Hai.

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Communicated by Jean-Pierre Crouzeix.

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Thuy, L.Q., Hai, T.N. A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications. J Optim Theory Appl 175, 411–431 (2017). https://doi.org/10.1007/s10957-017-1176-2

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  • DOI: https://doi.org/10.1007/s10957-017-1176-2

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