Skip to main content
SpringerLink
Log in

Existence of a Solution and Variational Principles for Vector Equilibrium Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we prove an existence result for a solution to the vector equilibrium problems. Then, we establish variational principles, that is, vector optimization formulations of set-valued maps for vector equilibrium problems. A perturbation function

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980.

    Google Scholar 

  2. Giannessi, F., Theorems of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, New York, NY, pp. 151–186, 1980.

    Google Scholar 

  3. Giannessi, F., Editor, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, Holland, 2000.

    Google Scholar 

  4. Chen, G. Y., Goh, C. J., and Yang, X. Q., On Gap Functions for Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 55–72, 2000.

    Google Scholar 

  5. Ansari, Q. H., Wong, N. C., and Yao, J. C., The Existence of Nonlinear Inequalities, Applied Mathematical Letters, Vol. 12, pp. 89–92, 1999.

    Google Scholar 

  6. Antipin, A. S., On Convergence of Proximal Methods to Fixed Points of Extremal Mappings and Estimates of Their Rate of Convergence, Computational Mathematics and Mathematical Physics, Vol. 35, pp. 539–551, 1995.

    Google Scholar 

  7. Aubin, J. P., L'Analyse Non Linéaire et Ses Motivations Économiques, Masson, Paris, France, 1984.

    Google Scholar 

  8. Bianchi, M., and Schaible, S., Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31–43, 1996.

    Google Scholar 

  9. Blum, E., and Oettli, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 63, pp. 123–145, 1994.

    Google Scholar 

  10. BŔezis, H., Nirenberg, L., and Stampacchia, G., A Remark on Ky Fan's Minimax Principle, Bolletino della Unione Matematica Italiana, Vol. 6, pp. 293–300, 1972.

    Google Scholar 

  11. Chadli, O., Chbani, Z., and Riahi, H., Recession Methods for Equilibrium Problems and Applications to Variational and Hemivariational Inequalities, Discrete and Continuous Dynamical Systems, Vol. 5, pp. 185–195, 1999.

    Google Scholar 

  12. Chadli, O., Chbani, Z., and Riahi, H., Equilibrium Problems and Noncoercive Variational Inequalities, Optimization, Vol. 49, pp. 1–12, 1999.

    Google Scholar 

  13. Chadli, O., Chbani, Z., and Riahi, H., Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 105, pp. 299–323, 2000.

    Google Scholar 

  14. Hadjisavvas, N., and Schaible, S., From Scalar to Vector Equilibrium Problems in the Quasimonotone Case, Journal of Optimization Theory and Applications, Vol. 96, pp. 297–309, 1998.

    Google Scholar 

  15. Hadjisavvas, N., and Schaible, S., Quasimonotonicity and Pseudomonotonicity in Variational Inequalities and Equilibrium Problems, Generalized Convexity, Generalized Monotonicity: Recent Results, Edited by J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, Kluwer Academic Publishers, Dordrecht, Holland, pp. 257–275, 1998.

    Google Scholar 

  16. Husain, T., and Tarafdar, E., Simultaneous Variational Inequalities, Minimization Problems, and Related Results, Mathematica Japonica, Vol. 39, pp. 221– 231, 1994.

  17. Konnov, I. V., A Generalized Approach to Finding a Stationary Point and the Solution of Related Problems, Computational Mathematics and Mathematical Physics, Vol. 36, pp. 585–593, 1996.

    Google Scholar 

  18. Tarafdar, E., and Yuan, G. X. Z., Generalized Variational Inequalities and Their Applications, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 30, pp. 4171–4181, 1997.

    Google Scholar 

  19. Yuan, G. X. Z., KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, NY, 1999.

    Google Scholar 

  20. Ansari, Q. H., Vector Equilibrium Problems and Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 1–15, 2000.

    Google Scholar 

  21. Ansari, Q. H., Oettli, W., and SchlÄger, D., A Generalization of Vectorial Equilibria, Mathematical Methods of Operations Research, Vol. 46, pp. 147–152, 1997.

    Google Scholar 

  22. Bianchi, M., Hadjisavvas, N., and Schaible, S., Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 527–542, 1997.

    Google Scholar 

  23. Konnov, I. V., Combined Relaxation Method for Solving Vector Equilibrium Problems, Russian Mathematics, Vol. 39, pp. 51–59, 1995.

    Google Scholar 

  24. Lee, G. M., Kim, D. S., and Lee, B. S., On Noncooperative Vector Equilibrium, Indian Journal of Pure and Applied Mathematics, Vol. 27, pp. 735–739, 1996.

    Google Scholar 

  25. Oettli, W., A Remark on Vector-Valued Equilibria and Generalized Monotonicity, Acta Mathematica Vietnamica, Vol. 22, pp. 213–221, 1997.

    Google Scholar 

  26. Oettli, W., and SchlÄger, D., Generalized Vectorial Equilibria and Generalized Monotonicity, Functional Analysis with Current Applications in Science, Technology, and Industries, Edited by M. Brokate and A. H. Siddiqi, Pitman Research Notes in Mathematics, Longman, Essex, England, Vol. 377, pp. 145–154, 1998.

    Google Scholar 

  27. Tan, N. X., and Tinh, P. N., On the Existence of Equilibrium Points of Vector Functions, Numerical Functional Analysis and Optimization, Vol. 19, pp. 141– 156, 1998.

  28. Auchmuty, G., Variational Principles for Variational Inequalities, Numerical Functional Analysis and Optimization, Vol. 10, pp. 863–874, 1989.

    Google Scholar 

  29. Auslender, A., Optimisation: Méthodes Numériques, Masson, Paris, France, 1976.

    Google Scholar 

  30. Hearn, D. W., The Gap Function of a Convex Program, Operations Research Letters, Vol. 1, pp. 67–71, 1982.

    Google Scholar 

  31. Blum, E., and Oettli, W., Variational Principles for Equilibrium Problems, Parametric Optimization and Related Topics III, Edited by J. Guddat et al., Peter Lang, Frankfurt am Main, Germany, pp. 79–88, 1993.

    Google Scholar 

  32. Ansari, Q. H., Konnov, I. V., and Yao, J. C., Characterizations of Solutions for Vector Equilibrium Problems, Journal of Optimization Theory and Applications (to appear).

  33. Jeykumar, V., Oettli, W., and Natividad, M., A Solvability Theorem for a Class of Quasiconvex Mappings with Applications to Optimization, Journal of Mathematical Analysis and Applications, Vol. 179, pp. 537–546, 1993.

    Google Scholar 

  34. Tanaka, T., Generalized Quasiconvexities, Cone Saddle Points, and Minimax Theorem for Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 81, pp. 355–377, 1994.

    Google Scholar 

  35. Fan, K., A Generalization of Tichonoff's Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961.

    Google Scholar 

  36. Chowdhury, M. S. R., and Tan, K. K., Generalized Variational Inequalities for Quasimonotone Operators and Applications, Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 45, pp. 25–54, 1997.

    Google Scholar 

  37. Corley, H. W., Existence and Lagrange Duality for Maximization of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 54, pp. 489–501, 1987.

    Google Scholar 

  38. Corley, H. W., Optimality Conditions for Maximization of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 58, pp. 1–10, 1988.

    Google Scholar 

  39. Li, Z. F., and Chen, G. Y., Lagrangian Multipliers, Saddle Points, and Duality in Vector Optimization of Set-Valued Maps, Journal of Mathematical Analysis and Applications, Vol. 215, pp. 297–316, 1997.

    Google Scholar 

  40. Lin, L. J., Optimization of Set-Valued Functions, Journal of Mathematical Analysis and Applications, Vol. 186, pp. 30–51, 1994.

    Google Scholar 

  41. Luc, D. T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, New York, NY, Vol. 319, 1989.

    Google Scholar 

Download references

Authors
  1. Q. H. Ansari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. V. Konnov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. C. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ansari, Q.H., Konnov, I.V. & Yao, J.C. Existence of a Solution and Variational Principles for Vector Equilibrium Problems. Journal of Optimization Theory and Applications 110, 481–492 (2001). https://doi.org/10.1023/A:1017581009670

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1017581009670

Search

Navigation