Skip to main content
SpringerLink
Log in

Superefficiency in Vector Optimization with Nearly Subconvexlike Set-Valued Maps

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

Abstract

In the framework of locally convex topological vector spaces, we establish a scalarization theorem, a Lagrange multiplier theorem and duality theorems for superefficiency in vector optimization involving nearly subconvexlike set-valued maps.

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. Kuhn, H., Tucker, A.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–491. University of California Press, Berkeley (1951)

    Google Scholar 

  2. Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  3. Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)

    Article  MATH  Google Scholar 

  4. Hartley, R.: On cone efficiency, cone convexity, and cone compactness. SIAM J. Appl. Math. 34, 211–222 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  5. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Borwein, J.M., Zhuang, D.: Superefficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Corley, H.W.: Existence and Lagrangian duality for maximizations of set-valued functions. J. Optim. Theory Appl. 54, 489–500 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Gong, X.H.: Connectedness of efficient solution sets for set-valued maps in normed spaces. J. Optim. Theory Appl. 83, 83–96 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, Z.F., Chen, G.Y.: Lagrange multipliers, saddle points and duality in vector optimization of set-valued maps. J. Math. Anal. Appl. 215, 297–316 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48, 1187–1200 (1998)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Rong, W.D., Wu, Y.N.: Characterizations of superefficiency in cone-convexlike vector optimization with set-valued maps. Math. Methods Oper. Res. 48, 247–258 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mehra, A.: Superefficiency in vector optimization with nearly convexlike set-valued maps. J. Math. Anal. Appl. 276, 815–832 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Cristescu, R.: Ordered Vector Spaces and Linear Operators. Abacus, Kent (1976)

    MATH  Google Scholar 

  19. Schaefer, H.H.: Topological Vector Spaces. Springer, New York (1971)

    Google Scholar 

  20. 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 

  21. Sach, P.H.: Nearly subconvexlike set-valued maps and vector optimization problems. J. Optim. Theory Appl. 119, 335–356 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Cheng, Y.H., Fu, W.T.: Strong efficiency in a locally convex space. Math. Methods Oper. Res. 50, 373–384 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zhuang, D.: Density results for proper efficiencies. SIAM J. Control Optim. 32, 51–58 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gong, X.H., Dong, H.B., Wang, S.Y.: Optimality conditions for proper efficient solutions of vector set-valued optimization. J. Math. Anal. Appl. 284, 332–350 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu, People’s Republic of China

    L. Y. Xia

  2. Department of Mathematics, Suzhou University, Suzhou, Jiangsu, People’s Republic of China

    J. H. Qiu

Authors
  1. L. Y. Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. H. Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. H. Qiu.

Additional information

Communicated by H.P. Benson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xia, L.Y., Qiu, J.H. Superefficiency in Vector Optimization with Nearly Subconvexlike Set-Valued Maps. J Optim Theory Appl 136, 125–137 (2008). https://doi.org/10.1007/s10957-007-9291-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-007-9291-0

Keywords

Search

Navigation