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Journal of Optimization Theory and Applications

, Volume 155, Issue 3, pp 840–854 | Cite as

Local Uniqueness of Solutions to Ky Fan Vector Inequalities using Approximations as Derivatives

  • P. Q. Khanh
  • L. T. Tung
Article

Abstract

We establish sufficient conditions for the local uniqueness of solutions to Ky Fan vector strong and weak inequalities. By using approximations as generalized derivatives, our results are valid even in cases where the maps involved in the problems suffer infinite discontinuity at the considered point. Corollaries and examples show that the results extend and improve existing ones in the literature.

Keywords

Local uniqueness Ky Fan vector strong and weak inequalities Approximations 

Notes

Acknowledgements

This work was supported by the National Foundation for Science and Technology Development of Vietnam. The final work on the paper of the first author was carried out during his stay at the Vietnam Institute for Advanced Study in Mathematics (VIASM) as a Visiting Professor. The authors would like to thank the Handling Editor and Anonymous Referees for their valuable remarks and suggestions, and for VIASM for the hospitality.

References

  1. 1.
    Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972) Google Scholar
  2. 2.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) zbMATHGoogle Scholar
  3. 3.
    Blum, B., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bianchi, M., Schaible, S.: Equilibrium problems under generalized convexity and generalized monotonicity. J. Glob. Optim. 30, 124–134 (2004) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hai, N.X., Khanh, P.Q.: Existence of solutions to general quasi-equilibrium problems and applications. J. Optim. Theory Appl. 133, 317–327 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Li, S.J., Teo, K.L., Yang, X.Q.: On generalized vector quasi-equilibrium problems. Pac. J. Optim. 3, 301–307 (2007) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hai, N.X., Khanh, P.Q., Quan, N.H.: On the existence of solutions to quasivariational inclusion problems. J. Glob. Optim. 45, 565–581 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution sets to parametric quasi-equilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Li, S.J., Li, X.B., Teo, K.L.: The Hölder continuity of solutions to generalized vector equilibrium. Eur. J. Oper. Res. 199, 334–338 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Anh, L.Q., Khanh, P.Q.: Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces. J. Glob. Optim. 37, 449–465 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Tuan, L.A., Lee, G.M., Sach, P.H.: Upper semicontinuity results for the solution mapping of a mixed parametric generalized vector quasiequilibrium problem with moving cones. J. Glob. Optim. 47, 639–660 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lignola, M.B., Morgan, J.: α-Well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints. J. Glob. Optim. 36, 439–459 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fang, Y.P., Hu, R., Huang, N.J.: Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints. Comput. Math. Appl. 55, 89–100 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Anh, L.Q., Khanh, P.Q., Van, D.T.M., Yao, J.C.: Well-posedness for vector quasiequilibria. Taiwan. J. Math. 13, 713–737 (2009) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ceng, L.C., Petrucel, A., Yao, J.C.: Iterative approaches to solving equilibrium problems and fixed-point problems of infinitely many nonexpansive mappings. J. Optim. Theory Appl. 143, 37–58 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Peng, J.W., Yao, J.C.: Some new extragradient-like methods for generalized equilibrium problems, fixed point problems and variational inequality problems. Optim. Methods Softw. 25, 677–698 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Martinez-Legaz, J.E., Sosa, W.: Duality for equilibrium problems. J. Glob. Optim. 35, 311–319 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Khanh, P.Q., Tung, N.M.: Optimality and duality for nonsmooth set-valued vector equilibrium problems. Submitted for publication Google Scholar
  19. 19.
    Mordukhovich, B.S.: Multiobjective optimization with equilibrium constraints. Math. Program., Ser. B 117, 331–354 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Mordukhovich, B.S.: Characterizations of linear suboptimality for mathematical programs with equilibriums constraints. Math. Program., Ser. B 120, 261–283 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Khanh, P.Q., Tung, L.T.: First and second-order optimality conditions using approximations for vector equilibrium problems with constraints. Submitted for publication Google Scholar
  22. 22.
    Cottle, R.W.: Nonlinear programs with positively bounded Jacobians. SIAM J. Appl. Math. 14, 147–158 (1966) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Cottle, R.W., Stone, R.E.: On the uniqueness of solutions to linear complementarity problems. Math. Program. 27, 191–213 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Kyparisis, J.: Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems. Math. Program. 36, 105–113 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Tawhid, M.A.: On the local uniqueness of solutions of variational inequalities under H-differentiability. J. Optim. Theory Appl. 113, 149–164 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Luc, D.T.: Fréchet approximate Jacobian and local uniqueness of solutions in variational inequalities. J. Math. Anal. Appl. 268, 629–646 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Luc, D.T., Noor, M.A.: Local uniqueness of solutions of general variational inequalities. J. Optim. Theory Appl. 117, 103–119 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Khanh, P.Q., Luc, D.T., Tuan, N.D.: Local uniqueness of solutions for equilibrium problem. Adv. Nonlinear Var. Inequal. 15, 127–145 (2006) MathSciNetGoogle Scholar
  29. 29.
    Jourani, A., Thibault, L.: Approximations and metric regularity in mathematical programming in Banach spaces. Math. Oper. Res. 18, 390–400 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Khanh, P.Q., Tuan, N.D.: First and second-order optimality conditions using approximations for nonsmooth vector optimization in Banach spaces. J. Optim. Theory Appl. 136, 238–265 (2006) MathSciNetGoogle Scholar
  31. 31.
    Khanh, P.Q., Tuan, N.D.: First and second-order approximations as derivatives of mappings in optimality conditions for nonsmooth vector optimization. Appl. Math. Optim. 58, 147–166 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Khanh, P.Q., Tuan, N.D.: Optimality conditions without continuity in multivalued optimization using approximations as generalized derivatives. In: Mishra, S.K. (ed.) Recent Contributions in Nonconvex Optimization, pp. 47–61. Springer, Berlin (2011) Google Scholar
  33. 33.
    Anh, L.Q., Khanh, P.Q.: On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems. J. Math. Anal. Appl. 321, 308–315 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Gowda, M.S., Ravindran, G.: Algebraic univalence theorems for nonsmooth functions. J. Math. Anal. Appl. 252, 917–935 (2000) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsInternational University of Hochiminh CityHo Chi Minh CityVietnam
  2. 2.Department of Mathematics, College of SciencesCantho UniversityCanthoVietnam

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