Skip to main content
SpringerLink
Log in

Higher-Order Optimality Conditions for Set-Valued Optimization

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

Abstract

This paper deals with higher-order optimality conditions of set-valued optimization problems. By virtue of the higher-order derivatives introduced in (Aubin and Frankowska, Set-Valued Analysis, Birkhäuser, Boston, [1990]) higher-order necessary and sufficient optimality conditions are obtained for a set-valued optimization problem whose constraint condition is determined by a fixed set. Higher-order Fritz John type necessary and sufficient optimality conditions are also obtained for a set-valued optimization problem whose constraint condition is determined by a set-valued map.

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., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  2. Jahn, J.: Vector Optimization–Theory, Applications, Extensions. Springer, Berlin (2004)

    Google Scholar 

  3. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)

    Google Scholar 

  4. Taa, A.: Optimality conditions for vector mathematical programming via a theorem of the alternative. J. Math. Anal. Appl. 233, 233–245 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Minami, H.: Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on Banach space. J. Optim. Theory Appl. 41, 451–461 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lin, J.G.: Maximal vectors and multiobjective optimization. J. Optim. Theory Appl. 18, 41–64 (1976)

    Article  MATH  Google Scholar 

  7. Censor, Y.: Pareto optimality in multiobjective problems. Appl. Math. Optim. 4, 41–59 (1977)

    Article  MathSciNet  Google Scholar 

  8. Aghezzaf, A., Hachimi, M.: Second-order optimality conditions in multiobjective optimization problems. J. Optim. Theory Appl. 102, 37–50 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jiménez, B., Novo, V.: Second order necessary conditions in set constrained differentiable vector optimization. Math. Methods Oper. Res. 58, 299–317 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49, 123–144 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Luc, D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99–111 (1991)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Li, S.J., Yang, X.Q., Chen, G.Y.: Nonconvex vector optimization of set-valued mappings. J. Math. Anal. Appl. 283, 337–350 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jahn, J., Khan, A.A.: Generalized contingent epiderivatives in set-valued optimization: optimality conditions. Numer. Funct. Anal. Optim. 23, 807–831 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Crespi, G.P., Ginchev, I., Rocca, M.: First-order optimality conditions in set-valued optimization. Math. Methods Oper. Res. 63, 87–106 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Khan, A.A., Raciti, F.: A multiplier rule in set-valued optimization. Bull. Aust. Math. Soc. 68, 93–100 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

  18. Bigi, G., Castellani, M.: K-epiderivatives for set-valued function and optimization. Math. Methods Oper. Res. 55, 401–412 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Science, Chongqing University, Chongqing, 400044, China

    S. J. Li

  2. Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U1987, Perth, WA, 6845, Australia

    K. L. Teo

  3. Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong

    X. Q. Yang

Authors
  1. S. J. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. L. Teo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. X. Q. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. J. Li.

Additional information

Communicated by S. Schaible.

This research was partially supported by the Postdoctoral Fellowship Scheme of the Hong Kong Polytechnic University and the National Natural Science Foundation of China (Grant Number 60574073) and Natural Science Foundation Project of CQ CSTC (Grant 2007BB6117).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, S.J., Teo, K.L. & Yang, X.Q. Higher-Order Optimality Conditions for Set-Valued Optimization. J Optim Theory Appl 137, 533–553 (2008). https://doi.org/10.1007/s10957-007-9345-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-007-9345-3

Keywords

Search

Navigation