Skip to main content
SpringerLink
Log in

An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems

  • Original Article
  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, the notion of gap functions is extended from scalar case to vector one. Then, gap functions and generalized functions for several kinds of vector equilibrium problems are shown. As an application, the dual problem of a class of optimization problems with a system of vector equilibrium constraints (in short, OP) is established, the concavity of the dual function, the weak duality of (OP) and the saddle point sufficient condition are derived by using generalized gap functions.

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. Ansari Q.H., Chan W.K. and Yang X.Q. (2004). The system of vector quasi-equilibrium problems with applications. J. Global Optim. 29: 45–57

    Article  Google Scholar 

  2. Ansari Q.H., Schaible S. and Yao J.C. (2002). The system of generalized vector equilibrium problems with applications.. J. Global Optim. 22: 3–16

    Article  Google Scholar 

  3. Ansari Q.H. and Yao J.C. (1999). An existence result for generalized vector equilibrium problem. Appl. Math. Lett. 12: 53–56

    Article  Google Scholar 

  4. Bianchi M., Hadjisavvas N. and Schaible S. (1997). Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92(3): 527–542

    Article  Google Scholar 

  5. Blum E. and Oettli W. (1994). From optimization and variational inequalities to equilibrium problems. The Math. Student 63: 123–145

    Google Scholar 

  6. Chadli O., Wong N.C. and Yao J.C. (2003). Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 117: 245–266

    Article  Google Scholar 

  7. Chen G.Y., Goh C.J. and Yang X.Q. (2000). On gap functions for vector variational inequalities. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibrium, pp 55–72. Kluwer Academic Publishers, Dordrecht, Boston, London

    Google Scholar 

  8. Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems 541, Springer-Verlag, Berlin, Heidelberg (2005)

  9. Chen G.Y. and Yang X.Q. (2002). Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112(1): 97–110

    Article  Google Scholar 

  10. Chen G.Y., Yang X.Q. and Yu H. (2005). A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Global Optim. 32(4): 451–466

    Article  Google Scholar 

  11. Ding X.P., Yao J.C. and Lin L.J. (2004). Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces. J. Math. Anal. Appl. 298: 398–410

    Article  Google Scholar 

  12. Fang Y.P. and Huang N.J. (2004). Existence results for systems of strong implicit vector variational inequalities. Acta Math. Hungar. 103: 265–277

    Article  Google Scholar 

  13. Fang Y.P. and Huang N.J. (2004). Vector equilibrium type problems with (S)+-conditions. Optimization 53: 269–279

    Article  Google Scholar 

  14. Flores-Bazán Y.P. (2003). Existence theory for finite-dimensional pseudomonotone equilibrium problems. Acta Appl. Math. 77: 249–297

    Article  Google Scholar 

  15. Gerth C. and Weidner P. (1990). Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67: 297–320

    Article  Google Scholar 

  16. Giannessi F. Theorem of alternative, quadratic programs, and complementarity problems. In: Cottle R.W., Giannessi F., Lions J.L. (ed.) Variational Inequality and Complementarity Problems, pp. 151–186. John Wiley and Sons, Chichester, England

  17. Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibrium. Kluwer Academic Publishers, Dordrecht, Boston, London (2000)

  18. Goh C.J. and Yang X.Q. (2002). Duality in Optimization and Variational Inequalities. Taylor and Francis, London

    Google Scholar 

  19. Göpfert A., Riahi H., Tammer C. and Zălinescu C. (2003). Variational Methods in Partially Ordered Spaces. Springer-Verlag, Berlin

    Google Scholar 

  20. Hadjisavvas N. and Schaible S. (1998). From scalar to vector equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 96(2): 297–309

    Article  Google Scholar 

  21. Huang N.J. and Gao C.J. (2003). Some generalized vector variational inequalities and complementarity problems for multivalued mappings. Appl. Math. Lett. 16: 1003–1010

    Article  Google Scholar 

  22. Huang N.J., Li J. and Thompson H.B. (2003). Implicit vector equilibrium problems with applications. Math. Comput. Modelling 37: 1343–1356

    Article  Google Scholar 

  23. Huang, N.J., Li, J., Yao, J.C.: Gap functions and existence of solutions for a system of vector equilibrium problems. J. Optim. Theory Appl. (in press)

  24. Horst R., Pardalos P.M. and Thoai N.V. (1995). Introduce to Global Optimization. Kluwer Academic Publishers, Dordrecht, Boston, London

    Google Scholar 

  25. Isac G., Bulavski V.A. and Kalashnikov V.V. (2002). Complementarity, Equilibrium, Efficiency and Economics. Kluwer Academic Publishers, Dordrecht, Boston, London

    Google Scholar 

  26. Li J. and He Z.Q. (2005). Gap functions and existence of solutions to generalized vector variational inequalities. Appl. Math. Lett. 18(9): 989–1000

    Article  Google Scholar 

  27. Li J. and Huang N.J. (2005). Implicit vector equilibrium problems via nonlinear scalarisation. Bulletin of the Australian Mathematical Society 72(1): 161–172

    Article  Google Scholar 

  28. Li J., Huang N.J. and Kim J.K. (2003). On implicit vector equilibrium problems. J. Math. Anal. Appl. 283: 501–512

    Article  Google Scholar 

  29. Li S.J., Teo K.L. and Yang X.Q. (2005). Generalized vector quasi-equilibrium problems. Math. Meth. Oper. Res. 61: 385–397

    Article  Google Scholar 

  30. Lin L.J. and Chen H.L. (2005). The study of KKM theorems with applications to vector equilibrium problems and implicit vector variational inequalities problems. J. Global Optim. 32: 135–157

    Article  Google Scholar 

  31. Mastroeni G. (2003). Gap functions for equilibrium problems. J. Global Optim. 27: 411–426

    Article  Google Scholar 

  32. Yang X.Q. (2003). On the gap functions of prevariational inequalities. J. Optim. Theory Appl. 116: 437–452

    Article  Google Scholar 

  33. Yang X.Q. and Yao J.C. (2002). Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115: 407–417

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Information, China West Normal University, Nanchong, Sichuan, 637002, P.R. China

    Jun Li

  2. Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, P.R. China

    Nan-Jing Huang

Authors
  1. Jun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nan-Jing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nan-Jing Huang.

Additional information

This work was supported by the National Natural Science Foundation of China (10671135) and the Applied Research Project of Sichuan Province (05JY029-009-1).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, J., Huang, NJ. An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems. J Glob Optim 39, 247–260 (2007). https://doi.org/10.1007/s10898-007-9137-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-007-9137-1

Keywords

2000 Mathematics Subject Classification

Search

Navigation