Skip to main content
SpringerLink
Log in

Second-Order Karush–Kuhn–Tucker Optimality Conditions for Set-Valued Optimization Subject to Mixed Constraints

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we propose the concept of second-order composed adjacent contingent derivatives for set-valued maps and hence discuss the included relationship to the second-order composed contingent derivatives. Under the appropriate conditions, Lipschitz properties of the first order derivatives and a chain rule for such second-order composed contingent derivatives are demonstrated. By virtue of second-order composed adjacent contingent derivatives and second-order composed contingent derivatives, we establish second-order Karush–Kuhn–Tucker sufficient and necessary optimality conditions for a set-valued optimization problem subject to mixed constraints. Applying a separation theorem for convex sets and cone-Aubin properties, we address stronger necessary optimality conditions and extend the second-order Kurcyusz–Robinson–Zowe regularity assumption for a problem with mixed constraints, in which the derivatives of the objective and the constraint functions are considered in separated ways. When the results regress to vector optimization, we also extend and improve some recent existing results.

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. Gfrerer, H.: On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs. SIAM J. Optim. 23, 632–665 (2013)

    Article  MathSciNet  Google Scholar 

  2. Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems. Math. Program. 41, 73–96 (1988)

    Article  MathSciNet  Google Scholar 

  3. Gutiérrez, C., Jiménez, B., Novo, N.: On second order Fritz John type optimality conditions in nonsmooth multiobjective programming. Math. Program. 123, 199–223 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Taa, A.: Second-order conditions for nonsmooth multiobjective optimization problems with inclusion constraints. J. Glob. Optim. 50, 271–291 (2011)

    Article  MathSciNet  Google Scholar 

  6. Ning, E., Song, W., Zhang, Y.: Second order sufficient optimality conditions in vector optimization. J. Glob. Optim. 54, 537–549 (2012)

    Article  MathSciNet  Google Scholar 

  7. Jiménez, B., Novo, V.: Second-order necessary conditions in set constrained differentiable vector optimization. Math. Methods Oper. Res. 58(2), 299–317 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Khan, A.A., Tammer, C.: Second-order optimality conditions in set-valued optimization via asymptotic derivatives. Optimization 62, 743–758 (2013)

    Article  MathSciNet  Google Scholar 

  10. Zhu, S.K., Li, S.J., Teo, K.L.: Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization. J. Glob. Optim. 58, 673–679 (2014)

    Article  MathSciNet  Google Scholar 

  11. Anh, N.L.H., Khanh, P.Q.: Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives. J. Glob. Optim. 56, 519–536 (2013)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Khanh, P.Q., Tuan, N.D.: Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsoomth vector optimization. J. Optim. Theory Appl. 139, 243–261 (2008)

    Article  MathSciNet  Google Scholar 

  14. Isac, G., Khan, A.A.: Second-order optimality conditions in set-valued optimization by a new tangential derivative. Acta Math. Vietnam. 34, 81–90 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Tuan, N.D.: First and second-order optimality conditions for nonsmooth vector optimization using set-valued directional derivatives. Appl. Math. Comput. 251, 300–317 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Götz, A., Jahn, J.: The Lagrange multiplier rule in set-valued optimization. SIAM J. Optim. 10(2), 331–344 (2000)

    Article  MathSciNet  Google Scholar 

  17. Peng, Z.H., Xu, Y.H.: New second-order tangent epiderivatives and applications to set-valued optimization. J. Optim. Theory Appl. 172(1), 128–140 (2017)

    Article  MathSciNet  Google Scholar 

  18. Khanh, P.Q., Tuan, N.D.: Second-order optimality conditions with the envelope-like effect for set-valued optimization. J. Optim. Theory Appl. 167, 68–90 (2015)

    Article  MathSciNet  Google Scholar 

  19. Khanh, P.Q., Tuan, N.D.: Second-order conditions for open-cone minimizers and firm minimizers in set-valued optimization subject to mixed constraints. J. Optim. Theory Appl. 171, 45–69 (2016)

    Article  MathSciNet  Google Scholar 

  20. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  21. Flores-Bazan, F., Jimenez, B.: Strict efficiency in set-valued optimization. SIAM J. Control Optim. 48, 881–908 (2009)

    Article  MathSciNet  Google Scholar 

  22. 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  Google Scholar 

  23. Hu, Y.D., Ling, C.: Connectedness of cone superefficient point sets in locally convex topological vector spaces. J. Optim. Theory Appl. 107(2), 433–446 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the Natural Science Foundations of China Nos. 71471140 and 11771058.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Zhenhua Peng & Zhongping Wan

Authors
  1. Zhenhua Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhongping Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenhua Peng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peng, Z., Wan, Z. Second-Order Karush–Kuhn–Tucker Optimality Conditions for Set-Valued Optimization Subject to Mixed Constraints. Results Math 73, 101 (2018). https://doi.org/10.1007/s00025-018-0865-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-018-0865-y

Mathematics Subject Classification

Keywords

Search

Navigation