Skip to main content
SpringerLink
Log in

Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces

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

Abstract

In real ordered linear spaces, an equivalent characterization of generalized cone subconvexlikeness of set-valued maps is firstly established. Secondly, under the assumption of generalized cone subconvexlikeness of set-valued maps, a scalarization theorem of set-valued optimization problems in the sense of ϵ-weak efficiency is obtained. Finally, by a scalarization approach, an existence theorem of ϵ-global properly efficient element of set-valued optimization problems is obtained. The results in this paper generalize and improve some known results in the literature.

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Yang, X.M., Yang, X.Q., Chen, G.Y.: Theorems of the alternative and optimization with set-valued maps. J. Optim. Theory Appl. 107, 627–640 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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  MathSciNet  MATH  Google Scholar 

  6. Sach, P.H.: New generalized convexity notion for set-valued maps and application to vector optimization. J. Optim. Theory Appl. 125, 157–179 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, Z.M.: The optimality conditions for vector optimization of set-valued maps. J. Math. Anal. Appl. 237, 413–424 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Adán, M., Novo, V.: Weak and proper efficiency in set-valued optimization on real linear spaces. J. Convex Anal. 14(2), 275–296 (2007)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Yu, G.L., Liu, S.Y.: Globally proper saddle point in ic-cone-convexlike set-valued optimization problems. Acta Math. Sin. 25(11), 1921–1928 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tiel, J.V.: Convex Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  12. Adán, M., Novo, V.: Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness. Eur. J. Oper. Res. 149, 641–653 (2003)

    Article  MATH  Google Scholar 

  13. Jahn, J.: Vector Optimization—Theory, Applications and Extensions, 2nd edn. Springer, Berlin (2011)

    MATH  Google Scholar 

  14. Jahn, J.: Scalarization in vector optimization. Math. Program. 29, 203–218 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, Y.W., Li, Z.M.: Optimality condition and Lagrangian multipliers of vector optimization with set-valued maps in linear spaces. Oper. Res. Trans. 5, 63–69 (2001)

    Google Scholar 

  17. Rong, W.D., Wu, Y.N.: ϵ-Weak minimal solutions of vector optimization problems with set-valued maps. J. Optim. Theory Appl. 106, 569–579 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhou, Z.A.: Benson proper efficiency for vector optimization of generalized subconvexlike set-valued maps in ordered linear spaces. J. Shanghai Univ. 14, 374–379 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, L.G., Liu, S.Y.: Saddle points and Lagrangian dual problems of vector-valued map. J. Syst. Sci. Math. Sci. 25, 398–405 (2005)

    MATH  Google Scholar 

  20. Li, Z.F., Wang, S.Y.: Lagrange multipliers and saddle points in multiobjective programming. J. Optim. Theory Appl. 83, 63–81 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chinaie, M., Zafarani, J.: Image space analysis and scalarization for ϵ-optimization of multifunctions. J. Optim. Theory Appl. doi:10.1007/s10957-010-9657-6

  22. Adán, M., Novo, V.: Proper efficiency in vector optimization on real linear spaces. J. Optim. Theory Appl. 121, 515–540 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Z.A. Zhou was supported by the National Natural Science Foundation of China (11126348), the Natural Science Foundation Project of CQ CSTC (CSTC, 2011jjA00022, CSTC, 2010BB2090) and the Science and Technology Project of Chongqing Municipal Education Commission (KJ110827).

J.W. Peng was supported by the National Natural Science Foundation of China (10831009, 11171363), the Natural Science Foundation Project of Chongqing (CSTC, 2009BB8240), the Special Fund of Chongqing Key Laboratory (CSTC, 2011KLORSE01) and the Project of the Third Batch Support Program for Excellent Talents of Chongqing City High Colleges.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Chongqing University of Technology, Chongqing, 400054, China

    Z. A. Zhou

  2. School of Mathematics, Chongqing Normal University, Chongqing, 400047, China

    J. W. Peng

Authors
  1. Z. A. Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. W. Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. W. Peng.

Additional information

Communicated by Jen-Chih Yao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, Z.A., Peng, J.W. Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces. J Optim Theory Appl 154, 830–841 (2012). https://doi.org/10.1007/s10957-012-0045-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0045-2

Keywords

Search

Navigation