Nothing Special   »   [go: up one dir, main page]

Skip to main content

Advertisement

Log in

On composite vector variational-like inequalities and vector optimization problems

  • Original Paper
  • Published:
Computational Management Science Aims and scope Submit manuscript

Abstract

In this paper, we introduce a class of (weak) composite vector variational-like inequality problems and establish its relationship with composite vector optimization problem. We also prove the relation of a vector critical point in composite vector optimization problem with its weak efficient point, under the assumption of composite pseudo invexity. Using KKM Lemma, we derive result for existence of solutions of composite vector variational-like inequality problem. Furthermore, we define a gap function for the composite vector variational-like inequality problem and finally, as an application, we study a system of composite vector optimization problems and system of vector variational-like inequality problems, whose solutions imply the solution of Nash equilibrium problem. Examples are provided to illustrate the derived results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  • Ansari QH (2012) Topics in nonlinear analysis and optimization. World Education, Delhi

    Google Scholar 

  • Askar SS, Tiwari A (2009) First-order optimality conditions and duality results for multi-objective optimisation problems. Ann Oper Res 172:277–289

    Article  MATH  MathSciNet  Google Scholar 

  • Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Stud 63:123–145

    MATH  MathSciNet  Google Scholar 

  • Chen B, Huang N-J (2012) Vector variational-like inequalities and vector optimization problems in Asplund spaces. Optim Lett 6:1513–1525

    Article  MATH  MathSciNet  Google Scholar 

  • Facchinei F, Kanzow C (2007) Generalized Nash equilibrium problems. 4OR 5:173–210

  • Giannessi F (1980) Theorems of the alternative quadratic programs and complementarity problems. In: Cottle RW, Giannessi F, Lions JL (eds) Variational inequalities and complementarity problems. Wiley, Chichester, pp 151–186

    Google Scholar 

  • Hanson MA (1981) On sufficiency of the KuhnTucker conditions. J Math Anal Appl 80:545–550

    Article  MATH  MathSciNet  Google Scholar 

  • Huang N-J, Li J, Thompson HB (2003) Implicit vector equilibrium problems with applications. Math Comput Model Dyn Syst 37:1343–1356

    Article  MATH  MathSciNet  Google Scholar 

  • Jabarootian T, Zafarani J (2008) Generalized vector variational-like inequalities. J Optim Theory Appl 136:15–30

    Article  MATH  MathSciNet  Google Scholar 

  • Jahn J (2004) Vector optimization: theory, applications and extensions. Springer, Berlin

    Book  Google Scholar 

  • Jayswal A, Choudhury S, Verma RU (2014) Exponential type vector variational-like inequalities and vector optimization problems with exponential type invexities. J Appl Math Comput 45:87–97

    Article  MATH  MathSciNet  Google Scholar 

  • Krawczyk J (2007) Numerical solutions to coupled-constraint (or generalised Nash) equilibrium problems. Comput Manag Sci 4:183–204

    Article  MATH  MathSciNet  Google Scholar 

  • Mehta M, Chaudhary M (2014) Strong vector variational like inequality problems with properly quasimonotone bifunctions. J Inequal Appl 2014:142

    Article  MathSciNet  Google Scholar 

  • Mohan SR, Neogy K (1995) On invex sets and preinvex functions. J Math Anal Appl 189:901–908

    Article  MATH  MathSciNet  Google Scholar 

  • Oveisiha M, Zafarani J (2012) Vector optimization problem and generalized convexity. J Glob Optim 52:29–43

    Article  MATH  MathSciNet  Google Scholar 

  • Padhan SK, Nahak C (2015) Second-order duality for invex composite optimization. J Egypt Math Soc 23:149–154

    Article  MathSciNet  Google Scholar 

  • Pang J-S, Fukushima M (2009) Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput Manag Sci 6:373–375

    Article  MATH  MathSciNet  Google Scholar 

  • Rockafellar RT (1988) First and second order epi-differentiation. Trans Am Math Soc 307:75–108

    Article  MATH  MathSciNet  Google Scholar 

  • Ruiz-Garzón-Lizana G, Osuna-Gómez R, Rufián-Lizana A (2004) Relationships between vector variational-like inequality and optimization problems. Eur J Oper Res 157:113–119

    Article  Google Scholar 

  • Sun XK, Li SJ (2013) Duality and gap function for generalized multivalued \(\epsilon \)-vector variational inequality. Appl Anal 92:482–492

    Article  MATH  MathSciNet  Google Scholar 

  • Verma RU (1998) Generalized variational inequalities and associated nonlinear equations. Czechoslov Math J 48:413–418

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anurag Jayswal.

Additional information

This research is financially supported by DST, New Delhi, India, through grant no. SR/FTP/MS-007/2011.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jayswal, A., Singh, S. & Choudhury, S. On composite vector variational-like inequalities and vector optimization problems. Comput Manag Sci 12, 577–594 (2015). https://doi.org/10.1007/s10287-015-0239-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10287-015-0239-9

Keywords

Mathematics Subject Classification

Navigation