Skip to main content
Log in

Generalized Minty vector variational-like inequalities and vector optimization problems in Asplund spaces

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We consider two generalized Minty vector variational-like inequalities and investigate the relations between their solutions and vector optimization problems for non-differentiable α-invex functions.

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

Access this article

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

  1. Al-Homidan S., Ansari Q.H.: Generalized minty vector variational-like inequalities and vector optimization problems. J. Optim. Theory Appl. 144, 1–11 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ansari Q.H., Lee G.M.: Nonsmooth vector optimization problems and minty vector variational inequalities. J. Optim. Theory Appl. 145, 1–16 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ansari, Q.H., Rezaie, M., Zafarani. J.: Generalized vector variational-like inequalities and vector optimization. J. Global Optim. (2012, in press)

  4. Chen, B., Huang, N.J.: Vector variational-like inequalities and vector optimization problems in Asplund spaces. Optim. Lett. (2012, in press)

  5. Chinchuluun A., Pardalos P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chinchuluun A., Pardalos P.M., Migdalas A., Pitsoulis L.: Pareto Optimality, Game Theory and Equilibria. Springer, New York (2008)

    Book  MATH  Google Scholar 

  7. Ceng L.C., Yao J.C.: On generalized variational-like inequalities with generalized monotone multivalued mappings. Appl. Math. Lett. 22, 428–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang Y.P., Hu R.: A non-smooth version of minty variational principle. Optimization 58, 401–412 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fakhar M., Zafarani J.: Generalized vector equilibrium problems for pseudomonotone bifunctions. J. Optim. Theory Appl. 126, 109–124 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Giannessi F.: Theorems on the alternative quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, Chichester (1980)

    Google Scholar 

  11. Giannessi F.: On Minty Variational Principle, New Trends in Mathematical Programming. Kluwer Academic, Dordrecht (1997)

    Google Scholar 

  12. Garzon G.R., Gomez R.O., Lizana A.R.: Generalized invex monotonicity. Eur. J. Oper. Res. 144, 501–512 (2003)

    Article  MATH  Google Scholar 

  13. Jabarootian T., Zafarani J.: Generalized invariant monotonicity and invexity of non-differentiable functions. J. Global Optim. 36, 537–564 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Long, X.J., Peng, J.W., Wu, S.Y.: Generalized vector variational-like inequalities and nonsmooth vector optimization problems. Optimization (2012, in press)

  16. Mordukhovich B.S.: Variational Analysis and Generalized Differential I, Basic theory, Grundlehren Ser., (Fundamental Principles of Mathematical Sciences). Springer, New York (2006)

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Polyak B.T.: Existence theorems and converegence of minimizing sequences in extremum problems with restrictions. Dokl. Akad. Nauk. USSR. 166, 287–290 (1966)

    Google Scholar 

  20. Rockafellar R.T., Wets R.J-B.: Variational Analysis. Springer, New York (1998)

    Book  MATH  Google Scholar 

  21. Rezaie M., Zafarani J.: Vector optimization and variational-like inequalities. J. Global Optim. 43, 47–66 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schirotzek W.: Nonsmooth Analysis. Springer, New York (2007)

    Book  MATH  Google Scholar 

  23. Soleimani-damaneh M.: Characterizations and applications of generalized invexity and monotonicity in Asplund spaces. Top. 18(1), 1–22 (2010)

    Article  MathSciNet  Google Scholar 

  24. Weir T., Mond B.: Preinvex functions in multiple-objective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang X.M., Yang X.Q.: Vector variational-like inequality with pseudoinvexity. Optimization 55, 157–170 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang X.M., Yang X.Q., Teo K.L.: Generalized invexity and generalized invariant monotonicity. J. Optim. Theory Appl. 117, 607–625 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang X.M., Yang X.Q., Teo K.L.: Some remarks on the minty vector variational inequality. J. Optim. Theory Appl. 121, 193–201 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang X.M., Yang X.Q., Teo K.L.: Criteria for generalized invex monotonicity. Eur. J. Oper. Res. 164, 115–119 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zopounidis C., Pardalos P.M.: Handbook of Multicriteria Analysis Series: Applied Optimization. Springer, New York (2010)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. Zafarani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Oveisiha, M., Zafarani, J. Generalized Minty vector variational-like inequalities and vector optimization problems in Asplund spaces. Optim Lett 7, 709–721 (2013). https://doi.org/10.1007/s11590-012-0454-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-012-0454-z

Keywords

Navigation