Skip to main content
SpringerLink
Log in

Levitin–Polyak well-posedness of vector equilibrium problems

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

In this paper, two types of Levitin–Polyak well-posedness of vector equilibrium problems with variable domination structures are investigated. Criteria and characterizations for two types of Levitin–Polyak well-posedness of vector equilibrium problems are shown. Moreover, by virtue of a gap function for vector equilibrium problems, the equivalent relations between the Levitin–Polyak well-posedness for an optimization problem and the Levitin–Polyak well-posedness for a vector equilibrium problem are obtained.

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

  • Ansari QH (2000) Vector equilibrium problems and vector variational inequalities. In: Giannessi F(eds) Vector variational inequalities and vector equilibria. Mathematical theories. Kluwer, Dordrecht, pp 1–16

    Google Scholar 

  • Ansari QH, Bazan FF (2003) Generalized vector quasi-equilibrium problems with applications. J Math Anal Appl 277: 246–256

    Article  MATH  MathSciNet  Google Scholar 

  • Aubin JP, Ekeland I (1984) Applied nonlinear analysis. Wiley, New York

    MATH  Google Scholar 

  • Bianchi M, Hadjisavvas N, Schaible S (1997) Vector equilibrium problems with generalized monotone bifunctions. J Optim Theory Appl 92: 531–546

    Article  MathSciNet  Google Scholar 

  • Chen GY, Yang XQ (2002) Characterizations of variable domination structures via nonlinear scalarization. J Optim Theory Appl 112(1): 97–110

    Article  MATH  MathSciNet  Google Scholar 

  • Chen GY, Goh CJ, Yang XQ (1999) Vector network equilibrium problems and nonlinear scalarization methods. Math Methods Oper Res 49: 239–253

    MATH  MathSciNet  Google Scholar 

  • Chen GY, Yang XQ, Yu H (2005) A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J Global Optim 32: 451–466

    Article  MATH  MathSciNet  Google Scholar 

  • Fang YP, Hu R, Huang NJ (2008) Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints. Comput Math Appl 55(1): 89–100

    Article  MathSciNet  Google Scholar 

  • Ferrentino R (2005) Pointwise well-posedness in vector optimization and variational inequalities. Working paper, Department of Economic Sciences and Statistics, University of Salerno-Fisciano

  • Furi M, Vignoli A (1970) About well-posed optimization problems for functionals in metric spaces. J Optim Theory Appl 5(3): 225–229

    Article  MATH  Google Scholar 

  • Giannessi F (ed) (2000) Vector variational inequalities and vector equilibria. Mathematical theories. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Huang XX, Yang XQ (2006) Generalized Levitin–Polyak well-posedness in constrained optimization. SIAM J Optim 17(1): 243–258

    Article  MATH  MathSciNet  Google Scholar 

  • Huang XX, Yang XQ (2007a) Levitin–Polyak well-posedness of constrained vector optimization problems. J Global Optim 37: 287–304

    Article  MATH  MathSciNet  Google Scholar 

  • Huang XX, Yang XQ (2007b) Levitin–Polyak well-posedness in generalized variational inequality problems with functional constraints. J Ind Manag Optim 3(4): 671–684

    MATH  MathSciNet  Google Scholar 

  • Konsulova AS, Revalski JP (1994) Constrained convex optimization problems well-posedness and stability. Numer Funct Anal Optim 15: 889–907

    Article  MATH  MathSciNet  Google Scholar 

  • Kuratowski C (1958) Topologie. Panstwowe Wydawnicto Naukowa, Warszawa, vol 1

  • Lai TC, Yao JC (1996) Existence results for VVIP. Appl Math Lett 9(3): 17–19

    Article  MATH  MathSciNet  Google Scholar 

  • Levitin ES, Polyak BT (1966) Convergence of minimizing sequences in conditional extremum problems. Sov Math Dokl 7: 764–767

    MATH  Google Scholar 

  • Li SJ, Yan H, Chen GY (2003) Differential and sensitivity properties of gap functions for vector variational inequalities. Math Methods Oper Res 57: 377–391

    MATH  MathSciNet  Google Scholar 

  • Li SJ, Teo KL, Yang XQ (2005) Generalized vector quasi-equilibrium problems. Math Methods Oper Res 61: 385–397

    Article  MATH  MathSciNet  Google Scholar 

  • Li SJ, Teo KL, Yang XQ, Wu SY (2006) Gap functions and existence of solutions to Generalized vector quasi-equilibrium problems. J Global Optim 34: 427–440

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Song W (2002) On generalized vector equilibrium problems. Appl Math Lett 12: 53–56

    Google Scholar 

  • Tykhonov AN (1966) On the stability of the functional optimization problem. USSR Comput Math Math Phys 6: 28–33

    Article  Google Scholar 

  • Ward DE, Lee GM (2002) On relations between vector optimization problems and vector variational inequalities. J Optim Theory Appl 113(3): 583–596

    Article  MATH  MathSciNet  Google Scholar 

  • Zolezzi T (1996) Extended well-posedness of optimization problems. J Optim Theory Appl 91(1): 257–266

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Sciences, Chongqing University, Chongqing, 400044, China

    S. J. Li & M. H. Li

Authors
  1. S. J. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. H. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. J. Li.

Additional information

This research was partially supported by the National Natural Science Foundation of China (Grant number: 60574073) and Natural Science Foundation Project of CQ CSTC (Grant number: 2007BB6117).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, S.J., Li, M.H. Levitin–Polyak well-posedness of vector equilibrium problems. Math Meth Oper Res 69, 125–140 (2009). https://doi.org/10.1007/s00186-008-0214-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-008-0214-0

Keywords

Search

Navigation