Skip to main content
SpringerLink
Log in

Optimality Condition for Local Efficient Solutions of Vector Equilibrium Problems via Convexificators and Applications

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Fritz John and Karush–Kuhn–Tucker necessary conditions for local efficient solutions of constrained vector equilibrium problems in Banach spaces in which those solutions are regular in the sense of Ioffe via convexificators are established. Under suitable assumptions on generalized convexity, sufficient conditions are derived. Some applications to constrained vector variational inequalities and constrained vector optimization problems are also given.

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. Daniele, P.: Lagrange multipliers and infinite-dimensional equilibrium problems. J. Glob. Optim. 40, 65–70 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities, image space analysis and separation. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 153–215. Kluwer, Dordrecht (2000)

    Chapter  Google Scholar 

  3. Gong, X.H.: Optimality conditions for efficient solution to the vector equilibrium problems with constraints. Taiwan. J. Math. 16, 1453–1473 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Gong, X.H.: Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342, 1455–1466 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gong, X.H.: Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal. 73, 3598–3612 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Luu, D.V., Hang, D.D.: On optimality conditions for vector variational inequalities. J. Math. Anal. Appl. 412, 792–804 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Luu, D.V., Hang, D.D.: Efficient solutions and optimality conditions for vector equilibrium problems. Math. Methods Oper. Res. 79, 163–177 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ma, B.C., Gong, X.H.: Optimality conditions for vector equilibrium problems in normed spaces. Optimization 60, 1441–1455 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Morgan, J., Romaniello, M.: Scalarization and Kuhn–Tucker-like conditions for weak vector generalized quasivariational inequalities. J. Optim. Theory Appl. 130, 309–316 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ward, D.E., Lee, G.M.: On relations between vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 113, 583–596 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yang, X.Q.: Continuous generalized convex functions and their characterizations. Optimization 54, 495–506 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Demyanov, V.F.: Convexification and concavification of a positively homogeneous function by the same family of linear functions, Universita di Pisa, Report 3, 208, 802 (1994)

  13. Demyanov, V.F., Jeyakumar, V.: Hunting for a smaller convex subdifferential. J. Glob. Optim. 10, 305–326 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jeyakumar, V., Luc, D.T.: Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 101, 599–621 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jeyakumar, V., Luc, D.T.: Approximate Jacobian matrices for continuous maps and \(C^1\)-optimization. SIAM J. Control Optim. 36, 1815–1832 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)

    MATH  Google Scholar 

  17. Michel, P., Penot, J.-P.: Calcul sous-différentiel pour des fonctions lipschitziennes et nonlipschitziennes. C. R. Math. Acad. Sci. 12, 269–272 (1984)

    MathSciNet  MATH  Google Scholar 

  18. Mordukhovich, B.S., Shao, Y.: On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal. 2, 211–228 (1995)

    MathSciNet  MATH  Google Scholar 

  19. Luu, D.V.: Convexificators and necessary conditions for efficiency. Optimization 63, 321–335 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luu, D.V.: Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 160, 510–526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Golestani, M., Nobakhtian, S.: Convexificators and strong Kuhn–Tucker conditions. Comp. Math. Appl. 64, 550–557 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ioffe, A.D.: Necessary and sufficient conditions for a local minimum. 1: a reduction theorem and first order conditions. SIAM J. Control Optim. 17, 245–250 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to the referees for their valuable comments and suggestions which improve the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2014.61.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Vietnam Academy of Science and Technology, Thang Long University, Hanoi, Vietnam

    Do Van Luu

Authors
  1. Do Van Luu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Do Van Luu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Luu, D. Optimality Condition for Local Efficient Solutions of Vector Equilibrium Problems via Convexificators and Applications. J Optim Theory Appl 171, 643–665 (2016). https://doi.org/10.1007/s10957-015-0815-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0815-8

Keywords

Mathematics Subject Classification

Search

Navigation