Skip to main content
SpringerLink
Log in

Robust duality for robust efficient solutions in uncertain vector optimization problems

  • Original Paper
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we introduce second-order subdifferentials of vector-valued maps and single-valued maps, respectively, and discuss some properties of the second-order (scalar) subdifferentials. In addition, we obtain a necessary and sufficient optimality condition of robust efficient solutions to the uncertain vector optimization problem. We also introduce a Wolfe type dual problem of the uncertain vector optimization problem. Finally, we establish robust weak duality and robust strong duality between the uncertain vector optimization problem and its robust dual problem.

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. Beck, A., Ben-Tal, A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 37, 1–6 (2009)

  2. Ben-Tal, A., Nemirovski, A.: Robust optimization methodology and applications. Math. Program. 92, 453–480 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jeyakumar, V., Li, G.Y.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20, 3384–3407 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jeyakumar, V., Li, G., Lee, G.M.: Robust duality for generalized convex programming problems under data uncertainty. Nonlinear Anal. 75, 1362–1373 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jeyakumar, V., Lee, G.M., Li, G.: Characterizing robust solution sets of convex programs under data uncertainty. J. Optim. Theory Appl. 164, 407–435 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Clarke, F.H.: Optimization and nonsmooth analysis. Society for Industrial and Applied Mathematics (1990)

  7. Ioffe, A.D.: Proximal analysis and approximate subdifferentials. J. Lond. Math. Soc. 2, 175–192 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yang, X.Q.: A Hahn–Banach theorem in ordered linear spaces and its applications. Optimization 25, 1–9 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Azimov, A.Y., Kasimov, R.N.: On weak conjugacy, weak subdifferentials and duality with zero gap in nonconvex optimization. Int. J. Appl. Math. 1, 171–192 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Kasimbeyli, R., İnceoğlu, G.: The properties of the weak subdifferentials. Gazi Univ. J. Sci. 23(1), 49–52 (2010)

    Google Scholar 

  12. Kasimbeyli, R., Mammadov, M.: On weak subdifferentials, directional derivatives, and radial epiderivatives for nonconvex functions. SIAM J. Optim. 20, 841–855 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheraghi, P., Farajzadeh, A.P., Milovanović, G.V.: Some notes on weak subdifferential. Filomat. 31, 3407–3420 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. İnceoğlu, G.: Some properties of second-order weak subdifferentials. Turk. J. Math. 45, 955–960 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Anh, N.L.H.: Mixed type duality for set-valued optimization problems via higher-order radial epiderivatives. Numer. Funct. Anal. Optim. 37, 823–838 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hernández, E., Rodríguez-Marín, L.: Weak and strong subgradients of set-valued maps. J. Optim. Theory Appl. 149, 352–365 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sun, X.K., Li, X.B., Long, X.J., et al.: On robust approximate optimal solutions for uncertain convex optimization and applications to multi-objective optimization. Pac. J. Optim. 13, 621–643 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Chuong, T.D., Kim, D.S.: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748–762 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chuong, T.D.: Optimality and duality for robust multiobjective optimization problems. Nonlinear Anal. 134, 127–143 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, J., Köbis, E., Yao, J.C.: Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J. Optim. Theory Appl. 181, 411–436 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Suneja, S.K., Khurana, S., Bhatia, M.: Optimality and duality in vector optimization involving generalized type I functions over cones. J. Glob. Optim. 49, 23–35 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lee, J.H., Jiao, L.: On quasi \(\epsilon \)-solution for robust convex optimization problems. Optim. Lett. 11, 1609–1622 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sun, X.K., Peng, Z.Y., Guo, X.L.: Some characterizations of robust efficient solutions for uncertain convex optimization problems. Optim. Lett. 10, 1463–1478 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sun, X.K., Teo, K.L., Long, X.J.: Some characterizations of approximate solutions for robust semi-infinite optimization problems. J. Optim. Theory Appl. 191, 281–310 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Huy, N.Q., Tuyen, N.V.: New second-order optimality conditions for a class of differentiable optimization problems. J. Optim. Theory Appl. 171, 27–44 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Boyd, S., Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  27. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the National Natural Science Foundation of China (No.11971078) and the Group Building Project for Scientifc Innovation for Universities in Chongqing (CXQT21021).

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Yuwen Zhai & Qilin Wang

  2. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

    Tian Tang

Authors
  1. Yuwen Zhai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qilin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tian Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qilin Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhai, Y., Wang, Q. & Tang, T. Robust duality for robust efficient solutions in uncertain vector optimization problems. Japan J. Indust. Appl. Math. 40, 907–928 (2023). https://doi.org/10.1007/s13160-022-00562-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-022-00562-7

Keywords

Mathematics Subject Classification

Search

Navigation