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Existence of Solutions and Algorithms for Bilevel Vector Equilibrium Problems: An Auxiliary Principle Technique

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Abstract

The main goal of this paper is to introduce and study bilevel vector equilibrium problems. We first establish some existence results for solutions of vector equilibrium problems and mixed vector equilibrium problems. We study the existence of solutions of bilevel vector equilibrium problems by considering a vector Thikhonov-type regularization procedure. By using this regularization procedure and existence results for mixed vector equilibrium problems, we establish some existence results for solutions of bilevel vector equilibrium problems. By using the auxiliary principle, we propose an algorithm for finding the approximate solutions of bilevel vector equilibrium problems. The strong convergence of the proposed algorithm is also studied.

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References

  1. Harker, P.T., Pang, J.-S.: Existence of optimal solutions to mathematcal programs with equilibrium constraints. Oper. Res. Lett. 7(2), 61–64 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bard, J.F.: Practical Bilevel Optimization: Applications and Algorithms. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  3. Dempe, S.: Foundations of Bilevel Programming. Kluwer Academie Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  4. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996)

    Book  MATH  Google Scholar 

  5. Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimizations with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  6. Bonnel, H., Morgan, J.: Semivectorial bilevel optimization problem: penalty approach. J. Optim. Theory Appl. 131, 365–382 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Liou, Y.C., Yang, X.Q., Yao, J.C.: Mathematical programs with vector optimization constraints. J. Optim. Theory Appl. 126(2), 345–355 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ansari, Q.H.: Vector equilibrium problems and vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 1–15. Kluwer Academic Publishers, Dordrecht (2000)

    Chapter  Google Scholar 

  9. Ansari, Q.H., Konnov, I.V., Yao, J.C.: Existence of a solution and variational principles for vector equilibrium problems. J. Optim. Theory Appl. 110, 481–492 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ansari, Q.H., Konnov, I.V., Yao, J.C.: Characterizations of solutions for vector equilibrium problems. J. Optim. Theory Appl. 113, 435–447 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ansari, Q.H., Oettli, W., Schläger, D.: A generalization of vectorial equilibria. Math. Methods Oper. Res. 46, 147–152 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Flores-Bazán, F., Flores-Bazán, F.: Vector equilibrium problems under asymptotic analysis. J. Glob. Optim. 26, 141–166 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hadjisavvas, N., Schaible, S.: From scalar to vector equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 96, 297–309 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kassay, G., Miholca, M.: Existence results for vector equilibrium problems given by a sum of two functions. J. Glob. Optim. 63, 195–211 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chadli, O., Chbani, Z., Riahi, H.: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J. Optim. Theory Appl. 105, 299–323 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dinh, B.V., Muu, L.D.: On penalty and gap function methods for bilevel equilibrium problems. J. Appl. Math. 2011, 1–14 (2011)

  19. Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. 74, 6121–6129 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Oliveira, P.R., Santos, P.S.M., Silva, A.N.: A Tikhonov-type regularization for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 401, 336–342 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ding, X.P.: Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces. J. Optim. Theory Appl. 146, 347–357 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cohen, G.: Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl. 32, 277–305 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cohen, G.: Auxiliary problem principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)

  25. Tanaka, T.: Generalized semicontinuity and existence theorems for cone saddle points. Appl. Math. Optim. 36, 313–322 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lalitha, C.S., Mehta, M.: Vector variational inequalities with cone-pseudomonotone bifunctions. Optimization 54, 327–338 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yu, S.J., Yao, J.C.: On vector variational inequalities. J. Optim. Theory Appl. 89, 749–769 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was funded by the National Plan for Science, Technology and Innovation (MAARIFAH)—King Abdulaziz City for Science and Technology—through the Science and Technology Unit at King Fahd University of Petroleum and Minerals (KFUPM)—the Kingdom of Saudi Arabia, Award Number 13-MAT1682-04.

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Authors and Affiliations

  1. Department of Economics, Faculty of Economics and Social Sciences, Ibn Zohr University, B.P. 8658, Poste Dakhla, Agadir, Morocco

    Ouayl Chadli

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Qamrul Hasan Ansari

  3. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Qamrul Hasan Ansari & Suliman Al-Homidan

Authors
  1. Ouayl Chadli
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  2. Qamrul Hasan Ansari
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  3. Suliman Al-Homidan
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Corresponding author

Correspondence to Qamrul Hasan Ansari.

Additional information

Communicated by I.V. Konnov.

Authors are grateful to the referees for their valuable comments and suggestions to improve the previous draft of the manuscript.

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Chadli, O., Ansari, Q.H. & Al-Homidan, S. Existence of Solutions and Algorithms for Bilevel Vector Equilibrium Problems: An Auxiliary Principle Technique. J Optim Theory Appl 172, 726–758 (2017). https://doi.org/10.1007/s10957-017-1062-y

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