Skip to main content
SpringerLink
Log in

Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems

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

Abstract

The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.

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. Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin L (ed.) Mathematical analysis and applications, Part A, Academic Press, New York, 1981, 160–229

    Google Scholar 

  2. Aubin, J.P., Frankowska, H. Set-Valued Analysis. Birkhäuser, Boston, 1990

    MATH  Google Scholar 

  3. Chen, G.Y., Jahn, J. Optimality conditions for set-valued optimization problems. Math. Meth. Oper. Res., 48: 187–200 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fu, J.Y., Wang, Y.H. Acrwise connected cone-convex functions and mathematical programming. J. Optim. Theory Appl., 118: 339–352 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jahn, J. Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin, Heidelberg, 2004

    Book  MATH  Google Scholar 

  6. Jahn, J., Khan, A.A., Zeilinger, P. Second-order optimality conditions in set optimization. J. Optim. Theory Appl., 125: 331–347 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jahn, J., Rauh, R. Contingent epiderivatives and set-valued optimization. Math. Meth. Oper. Res., 46: 193–211 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lalitha, C.S., Dutta, J., Govll, M.G. Optimality criteria in set-valued optimization. J. Aust. Math. Soc., 75: 221–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, S.J., Zhu, S.K., Teo, K.L. New generalized second-order contingent epiderivatives and set-valued optimization problems. J. Optim. Theory Appl., 152: 587–604 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. 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 

  12. Xu, Y.H. Optimality Conditions for Set-valued Optimization Problems. Doctoral thesis, Xidian University, 2003

    Google Scholar 

  13. Xu, Y.H., Song, X.S. Relationship between ic-cone-convexness and nearly cone-subconvexlikeness. Appl. Math. Lett., 24: 1622–1624 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. 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 

  16. Yu, G.L. Optimality of global proper efficiency for cone-arcwise connected set-valued optimization using contingent epiderivative. Asia-Pac. J. Oper. Res., 30(3): 1340004-1–1340004-10 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yu, G.L., Liu, S.Y. Optimality conditions of global proper efficient solutions for set-valued optimization problem. Acta Math. Appl. Sin., 33: 150–160 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Zhu, S.K., Li, S.J., Teo, K.L. Second-order Karush-Kuhn-Tucker optimality conditions for set-valued optimization. J. Glob. Optim., 58: 673–692 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanchang University, Nanchang, 330031, China

    Zhen-hua Peng & Yi-hong Xu

Authors
  1. Zhen-hua Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi-hong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi-hong Xu.

Additional information

Supported by the National Natural Science Foundation of China Grant 11461044, the Natural Science Foundation of Jiangxi Province (20151BAB201027) and the Science and Technology Foundation of the Education Department of Jiangxi Province(GJJ12010).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peng, Zh., Xu, Yh. Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems. Acta Math. Appl. Sin. Engl. Ser. 34, 183–196 (2018). https://doi.org/10.1007/s10255-018-0738-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-018-0738-x

Keywords

2000 MR Subject Classification

Search

Navigation