Skip to main content
SpringerLink
Log in

A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems

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

Abstract

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. In this paper we introduce a nonlinear scalarization function for a variable domination structure. Several important properties, such as subadditiveness and continuity, of this nonlinear scalarization function are established. This nonlinear scalarization function is applied to study the existence of solutions for generalized quasi-vector equilibrium problems.

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. J.P. Aubin I. Ekeland (1984) Applied Nonlinear Analysis J. Wiley New York

    Google Scholar 

  2. J.P. Aubin H. Frankowska (1990) Set-valued Analysis Birkhauser Boston, Berlin

    Google Scholar 

  3. M. Bianchi N. Hadjisavvas S. Schaible (1999) ArticleTitleVector equilibrium problems with generalized monotone bifunctions Journal of Optimization Theory and Applications 92 531–546

    Google Scholar 

  4. M. Bianchi S. Schaible (1996) ArticleTitleGeneralized monotone bifunctions and equilibrium problems Journal of Optimization Theory and Application 90 31–43 Occurrence Handle10.1007/BF02192244

    Article  Google Scholar 

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

    Google Scholar 

  6. G.Y. Chen (1992) ArticleTitleExsitence of solutions for a vector variational inequality: an extension of the Hartmann-Stampachia theorem Journal of Optimization Theory and Applications 74 445–456 Occurrence Handle10.1007/BF00940320

    Article  Google Scholar 

  7. G.Y. Chen S.J. Li (1996) ArticleTitleExistence of solutions for generalized vector quasi-variational inequality Journal of Optimization Theory and Applications 90 321–334 Occurrence Handle10.1007/BF02190001

    Article  Google Scholar 

  8. G.Y. Chen X.Q. Yang (2002) ArticleTitleCharacterizations of variable domination structures via a nonlinear scalarization Journal of Optimizaton Theory and Applications 112 97–110 Occurrence Handle10.1023/A:1013044529035

    Article  Google Scholar 

  9. B.C. Evans (1971) ArticleTitleOn the basic theorem of complementarity Mathematical Programming 1 68–75 Occurrence Handle10.1007/BF01584073

    Article  Google Scholar 

  10. Gerth(Tammer), Chr. and Weidner, P. (1990), Nonconvex secparation theorems and some applications in vector optimization, Journal of Optimization Theory and Applcaiotns, 67, 297–320.

    Google Scholar 

  11. F. Giannessi (1980) Theorems of alternative, quadratic programs and complementarity problems R.W. Cottle (Eds) et al. Variational Inequality Complementary Problems Wiley New York 151–186

    Google Scholar 

  12. V. Konnov J.C. Yao (1999) ArticleTitleExistence of solutions for generalized vector equilibrium problems Journal of Mathematical Analysis and Applications 23 328–335 Occurrence Handle10.1006/jmaa.1999.6312

    Article  Google Scholar 

  13. G.M. Lee D.S. Kim B.S. Lee (1996) ArticleTitleOn noncooperative vector equilibrium Indian Journal of Pure and Applied Math 27 735–739

    Google Scholar 

  14. L.J. Lin (1996) ArticleTitlePre-vector variational inequalities Bull. Austral. Math. Soc. 53 63–70

    Google Scholar 

  15. D.T. Luc (1989) Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems Springer-Velag Berlin 319

    Google Scholar 

  16. W. Oettli D. Schlager (1997) Generalized vectorial equilibria and generalized monotonicity M. Brokate (Eds) et al. Functional Analysis and Current Applications Longmang London 145–154

    Google Scholar 

  17. W. Oettli D. Schlager (1998) ArticleTitleExistence of equilibria for multivalued mappings Math. Meth. Oper. Res. 48 219–228 Occurrence Handle10.1007/s001860050024

    Article  Google Scholar 

  18. A.H. Siddiqi Q.H. Ansari R. Ahmad (1995) ArticleTitleOn vector variational inequality Journal of Optimization Theory and Applications 84 171–180 Occurrence Handle10.1007/BF02191741

    Article  Google Scholar 

  19. T. Tanino Y. Sawaragi (1980) ArticleTitleStability of nondominated solutions in multicriteria decision making Journal Optimization Theory and Applications 30 229–253 Occurrence Handle10.1007/BF00934497

    Article  Google Scholar 

  20. X.Q. Yang (1993) ArticleTitleVector variational inequalities and its duality Nonlinear Analysis – Theory, Method and Applications 21 867–877

    Google Scholar 

  21. X.Q. Yang C.J. Goh (1997) ArticleTitleOn vector variational inequality application to vector equilibria Journal of Optimization Theory and Applicaitons 95 431–443 Occurrence Handle10.1023/A:1022647607947

    Article  Google Scholar 

  22. P.L. Yu (1985) Multiple-criteria Decision Making: Concepts, Technieques, and Extensions Plenum Press New York, N.Y.

    Google Scholar 

  23. V. Jeyakumar W. Oettli M. Natividad (1993) ArticleTitleA solvability theorem for a class of quasiconvex mappings with applications to optimization Journal of Mathematical Analysis and Applications 179 537–546 Occurrence Handle10.1006/jmaa.1993.1368

    Article  Google Scholar 

  24. J.Y. Fu (2000) ArticleTitleGeneralized vector quasi-equilibrium problems Math. Meth. Oper. Res. 52 57–64 Occurrence Handle10.1007/s001860000058

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Systems Science, Chinese Academy of Sciences, Beijing, 100080, China

    G. Y. Chen

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    X. Q. Yang

  3. School of Economics and Management, Tsinghua University, Beijing, 100081, China

    H. Yu

Authors
  1. G. Y. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. X. Q. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Y. Chen.

Additional information

This paper is dedicated to Professor Franco Giannessi for his 68th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, G.Y., Yang, X.Q. & Yu, H. A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems. J Glob Optim 32, 451–466 (2005). https://doi.org/10.1007/s10898-003-2683-2

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-003-2683-2

Keywords

Mathematics Subject Classifications

Search

Navigation