Skip to main content
SpringerLink
Log in

Henig Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions

  • Original Papers
  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we study Henig efficiency in vector optimization with nearly cone-subconvexlike set-valued function. The existence of Henig efficient point is proved and characterization of Henig efficiency is established using the method of Lagrangian multiplier. As an interesting application of the results in this paper, we establish a Lagrange multiplier theorem for super efficiency in vector optimization with nearly conesubconvexlike set-valued function.

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. Aubin, J.P., Cellina A. Differential inclusions: Set-valued maps and viability theory. Springer Verlag, Berlin, Germany, 1984

  2. Aubin, J.P., Frankowska, H. Set-Valued Analysis. Birkhauser, Basel, Switzerland, 1990

  3. Benson, H.P. An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl., 71: 232–241 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bowein, J.M., Zhuang, D.M. Super efficiency in vector optimization. Trans. Amer. Math. Soc., 338: 105–122 (1993)

    Article  MathSciNet  Google Scholar 

  5. Bowein, J.M., Zhuang, D.M. Super efficiency in convex vector optimization. Math. Methods Oper. Res., 35: 175–184 (1991)

    Article  Google Scholar 

  6. Chen, G.Y., Jahn, J. Special Issue on Set-Valued Optimization. Math. Methods Oper. Res., 48: 151–285 (1998)

    Article  Google Scholar 

  7. Corley, H.W. Existence and lagrange duality for maximization of set-valued functions. J. Optim. Theory Appl., 54: 489–501 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Corley, H.W. Optimality conditions for maximizations of set-valued functions. J. Optim. Theory Appl., 58: 1–10 (1988)

    Article  MathSciNet  Google Scholar 

  9. Ferrero, O. Theorems of the alternative for set-valued functions in infinite- dimensional spaces. Optimization, 20: 167–175 (1989)

    MATH  MathSciNet  Google Scholar 

  10. Gong, X.H. Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl., 307: 12–31 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Guerraggio, A., Molho, E., Zaffaroni, A. On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl., 82: 1–21 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Henig, M.I. Proper efficiency with respect to cones. J. Optim. Theory Appl., 36: 387–407 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jahn, J. Mathematical vector optimization in partially order linear spaces. Verlag Peter Lang, Frankfurt, 1986

  14. Klein, K., Thompson, A.C. Theory of correspondences. John Wiley, New York, 1984

  15. Li, Z. A theorem of the alternative and its application to the optimization of set-valued maps. J. Optim. Theory Appl., 100: 365–375 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Li, Z.F. Benson proper efficiency in the vector optimization of set-valued maps. J. Optim. Theory Appl., 98: 623–649 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lin, L.J. Optimization of set-valued functions. J. Math. Anal. Appl., 186: 30–51 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mehra, A. Super efficiency in vector optimization with neraly convexlike set-valued maps. J. Math. Anal. Appl., 276: 815–832 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Paeck, S. Convexlike and concavelike conditions in alternative, minimax, and minimization theorems. J. Optim. Theory Appl., 14: 317–332 (1992)

    Article  MathSciNet  Google Scholar 

  20. Qiu, Q.S., Fu, W.T. The connectedness of super efficient solution sets of the optimization problem for set-valued mapping. J. Sys. Sci. Math. Sci., 22: 107–114 (2002) (in Chinese)

    MATH  Google Scholar 

  21. Rong W.D., Wu, Y.N. Characterization of super efficiency in cone-convex-like vector optimization with set-valued maps. Math. Methods Oper. Res., 48: 247–258 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Song W. Lagrangian duality for minimization of nonconvex multifunctions. J. Optim. Theory Appl., 93: 167–182 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tanaka T., Kuroiwa D. Some general conditions assuring intA + B =int(A + B). Applied Mathematics Letters, 6: 51–53 (1993)

    Article  MATH  Google Scholar 

  24. Yang X.M., Li, D., Wang, S.Y. Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions. J. Optim. Theory Appl., 110: 413–427 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zheng X.Y. The domination property for efficiency in locally convex spaces. J. Math. Anal. Appl., 213: 455–467 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zheng X.Y. Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl., 94: 469–486 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Qiu-sheng Qiu

  2. Department of Mathematics, Shanghai University, Shanghai, 200436, China

    Qiu-sheng Qiu

Authors
  1. Qiu-sheng Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiu-sheng Qiu.

Additional information

Supported by the Natural Science Foundation of Zhejiang Province, China (M103089).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Qiu, Qs. Henig Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions. Acta Mathematicae Applicatae Sinica, English Series 23, 319–328 (2007). https://doi.org/10.1007/s10255-007-0374-3

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-007-0374-3

Keywords

2000 MR Subject Classification

Search

Navigation