Skip to main content
SpringerLink
Log in

Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems

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

Abstract

This paper is devoted to the Levitin–Polyak well-posedness by perturbations for a class of general systems of set-valued vector quasi-equilibrium problems (SSVQEP) in Hausdorff topological vector spaces. Existence of solution for the system of set-valued vector quasi-equilibrium problem with respect to a parameter (PSSVQEP) and its dual problem are established. Some sufficient and necessary conditions for the Levitin–Polyak well-posedness by perturbations are derived by the method of continuous selection. We also explore the relationships among these Levitin–Polyak well-posedness by perturbations, the existence and uniqueness of solution to (SSVQEP). By virtue of the nonlinear scalarization technique, a parametric gap function g for (PSSVQEP) is introduced, which is distinct from that of Peng (J Glob Optim 52:779–795, 2012). The continuity of the parametric gap function g is proved. Finally, the relations between these Levitin–Polyak well-posedness by perturbations of (SSVQEP) and that of a corresponding minimization problem with functional constraints are also established under quite mild assumptions.

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, Schaible S, Yao JC (2000) System of vector equilibrium problems and its applications. J Optim Theory Appl 107: 547–557

    Article  MathSciNet  MATH  Google Scholar 

  • Ansari QH, Schaible S, Yao JC (2002) The system of generalized vector equilibrium problems with applications. J Glob Optim 22: 3–16

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Berge C (1963) Topological spaces. Oliver and Boyd, London

    MATH  Google Scholar 

  • Ceng LC, Hadjisavvas N, Schaible S, Yao JC (2008) Well-posedness for mixed quasivariational-like inequalities. J Optim Theory Appl 139: 109–125

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Z (1988) Continuous selections of set-valued mappings and fixed point theorems. Acta Math Sinica (Chinese) 31: 456–463

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Chen JW, Wan Z (2011) Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces. J Inequal Appl 2011: 1–18

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Chen GY, Goh CJ, Yang XQ (2000) On gap functions for vector variational inequalities. In: Giannessi F. (ed) Vector variational inequalities and vector equilibria. Kluwer Academic, Dordrecht, pp 55–72

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Chen GY, Huang XX, Yang XQ (2005b) Vector optimization: set-valued and variational analysis. In: Lecture notes in economics and mathematical systems, vol 285. Springer, Berlin, pp 408–416

  • Chen JW, Cho YJ, Wan Z (2011) Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space. Fixed Point Theory Appl 2011: 91

    Article  MathSciNet  Google Scholar 

  • Chen JW, Wan Z, Zou YZ (2012) Strong convergence theorems for firmly nonexpansive-type mappings and equilibrium problems in Banach spaces. Optimization doi:10.1080/02331934.2011.626779

  • Ding XP, Kim WK, Tan KK (1992) A selection theorem an its applications. Bull Austral Math Soc 46: 205–212

    Article  MathSciNet  MATH  Google Scholar 

  • Fang M, Huang NJ, Kim J (2006) Existence results for systems of vector equilibrium problems. J Glob Optim 35: 71–83

    Article  MathSciNet  MATH  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: 89–100

    Article  MathSciNet  MATH  Google Scholar 

  • Fang YP, Huang NJ, Yao JC (2010) Well-posedness by perturbations of mixed variational inequalities in Banach spaces. Eur J Oper Res 201: 682–692

    Article  MathSciNet  MATH  Google Scholar 

  • Fu JY (2005) Vector equilibrium problems existence theorems and convexity of solution set. J Glob Optim 31: 109–119

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Giannessi F (1998) On Minty variational principle. In: Giannessi F, Komloski S, Tapcsack T (eds) New trends in mathematical programming. Kluwer Academic Publisher, Dordrech, pp 93–99

    Google Scholar 

  • Giannessi F (2000) Vector variational inequalities and vector equilibria: mathematical theories. Kluwer Academic, Dordrecht

    Book  MATH  Google Scholar 

  • Gutev VG (1998) Continuous selections for continuous set-valued mappings and finite-dimensional sets. Set Valued Anal 6: 149–170

    Article  MathSciNet  MATH  Google Scholar 

  • Homidan SA, Ansari QH, Schaible S (2007) Existence of solutions of systems of generalized implicit vector variational inequalities. J Optim Theory Appl 134: 515–531

    Article  MathSciNet  MATH  Google Scholar 

  • Huang NJ, Li J, Yao JC (2007) Gap functions and existence of solutions for a system of vector equilibrium problems. J Optim Theory Appl 133: 201–212

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Huang XX, Yang XQ (2010) Levitin–Polyak well-posedness of vector variational inequality problems with functional constraints. Numer Funct Anal Optim 31: 671–684

    Article  Google Scholar 

  • Huang XX, Yang XQ, Zhu DL (2009) Levitin–Polyak well-posedness of variational inequality problems with functional constraints. J Glob Optim 2: 159–174

    Article  MathSciNet  Google Scholar 

  • Hu R, Fang YP, Huang NJ (2010a) Levitin–Polyak well-posedness for variational inequalities and for optimization problems with variational inequalities. J Ind Manag Optim 6: 465–481

    Article  MathSciNet  MATH  Google Scholar 

  • Hu R, Fang YP, Huang NJ, Wong MM (2010b) Well-posedness of systems of equilibrium problems. Taiwan J Math 14: 2435–2446

    MathSciNet  MATH  Google Scholar 

  • Jiang B, Zhang J, Huang XX (2009) Levitin–Polyak well-posedness of generalized quasivariational inequalities with functional constraints. Nonlinear Anal 70: 1492–1530

    Article  MathSciNet  MATH  Google Scholar 

  • Konsulova AS, Revalski JP (1994) Constrained convex optimization problems well-posedness and stability. Numer Funct Anal Optim 7(8): 889–907

    Article  MathSciNet  Google Scholar 

  • Kuratowski K (1968) Topology, vol. 1 and 2. Academic Press, New York

    Google Scholar 

  • Lalitha CS, Bhatia G (2010) Well-posedness for parametric quasivariational inequality problems and for optimizations problems with quasivariational inequality constraints. Optimization 59: 997–1011

    Article  MathSciNet  MATH  Google Scholar 

  • Lemaire B, Ould Ahmed Salem C, Revalski JP (2002) Well-posedness by perturbations of variational inequalities. J Optim Theory Appl 115: 345–368

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Lignola MB, Morgan J (2000) Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J Glob Optim 16: 57–67

    Article  MathSciNet  MATH  Google Scholar 

  • Lignola MB, Morgan J (2001) Approximating solutions and α- well-posedness for variational inequalities and Nash equilibria. In: Decision and control in management science, Kluwer Academic Publishers, pp 367–378

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

    Article  MathSciNet  MATH  Google Scholar 

  • Li SJ, Li MH (2009) Levitin–Polyak well-posedness of vector equilibrium problems. Math Meth Oper Res 69: 125–140

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Lucchetti R, Patrone F (1981) A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities. Numer Funct Anal Optim 3: 461–476

    Article  MathSciNet  MATH  Google Scholar 

  • Peng JW, Wu SY (2010) The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems. Optim Lett 4: 501–512

    Article  MathSciNet  MATH  Google Scholar 

  • Peng JW, Wang Y, Zhao LJ (2009) Generalized Levitin–Polyak well-posedness of vector equilibrium problems. Fixed Point Theory Appl 2009: 1–14

    Article  MathSciNet  Google Scholar 

  • Peng JW, Wu SY, Wang Y (2012) Levitin–Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints. J Glob Optim 52: 779–795

    Article  MathSciNet  MATH  Google Scholar 

  • Tykhonov AN (1966) On the stability of the functional optimization problem. USSR J Comput Math Math Phys 6: 631–634

    Google Scholar 

  • Xu Z, Zhu DL, Huang XX (2008) Levitin–Polyak well-posedness in generalized vector variational inequality problem with functional constraints. Math Meth Oper Res 67: 505–524

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Jia-Wei Chen & Zhongping Wan

  2. School of Information and Mathematics, Yangtze University, Jingzhou, 434023, People’s Republic of China

    Jia-Wei Chen & Zhongping Wan

  3. Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University, Chinju, 660-701, Korea

    Yeol Je Cho

Authors
  1. Jia-Wei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhongping Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yeol Je Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jia-Wei Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, JW., Wan, Z. & Cho, Y.J. Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems. Math Meth Oper Res 77, 33–64 (2013). https://doi.org/10.1007/s00186-012-0414-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-012-0414-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation