Skip to main content
SpringerLink
Log in

Directional Compactness, Approximations and Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this article, we introduce and study a natural version of the directional compactness, which can be viewed as one of the effective tools for constructing sufficient conditions in nonsmooth vector equilibrium problems. We also provide the generalized Hadamard directional derivative notion which is closely related to a version of contingent derivative. The relation among the first-order approximations/the Clarke generalized Jacobian and the generalized Hadamard directional differentiability is formulated. Using the tool of approximations, a new version of the constraint qualification of the (CQ) type is proposed for establishing KT-type necessary nonsmooth optimality conditions via the generalized Hadamard directional derivatives for the weak/and strict efficiency of constrained nonsmooth vector equilibrium problems. As applications, we study a nonsmooth vector equilibrium problem with set, cone constraints using approximations and the constraint qualification (CQ).

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. Birkhauser, Boston (1990)

    Google Scholar 

  2. Allali, K., Amahroq, T.: Second order approximations and primal and dual necessary optimality conditions. Optimization 40, 229–246 (1997)

    Article  MathSciNet  Google Scholar 

  3. Auslender, A.: Stability in mathematical programming with nondifferentiable data. SIAM J. Control. Optim. 22, 239–254 (1984)

    Article  MathSciNet  Google Scholar 

  4. Flores-Bazán, F., Jiménez, B.: Strict efficiency in set-valued optimization. SIAM J. Control. Optim. 48(2), 881–908 (2009)

    Article  MathSciNet  Google Scholar 

  5. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 127–149 (1994)

    MathSciNet  Google Scholar 

  6. Bonnans, F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  Google Scholar 

  7. Bigi, G., Castellani, M.: Second order optimality conditions for differentiable multiobjective problems. RAIRO Oper Res. 34, 411–426 (2000)

    Article  MathSciNet  Google Scholar 

  8. Bigi, G.: On sufficient second order optimality conditions in multiobjective optimization. Math. Methods Oper. Res. 63, 77–85 (2006)

    Article  MathSciNet  Google Scholar 

  9. Ben-Tal, A.: Second-order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31(2), 143–165 (1980)

    Article  MathSciNet  Google Scholar 

  10. Constantin, E.: Necessary conditions for weak efficiency for nonsmooth degenerate multiobjective optimization problems. J. Global Optim. 75(1), 111–129 (2019)

    Article  MathSciNet  Google Scholar 

  11. Constantin, E.: Necessary conditions for weak minima and for strict minima of order two in nonsmooth constrained multiobjective optimization. J. Global Optim. 80(1), 1–17 (2021)

    Article  MathSciNet  Google Scholar 

  12. Constantin, E.: Second-order optimality conditions in locally Lipschitz inequality-constrained multiobjective optimization. J. Optim. Theory Appl. 186, 50–67 (2020)

    Article  MathSciNet  Google Scholar 

  13. Constantin, E.: First-order necessary conditions in locally Lipschitz multiobjective optimization. Optimization 67(9), 1447–1460 (2018)

    Article  MathSciNet  Google Scholar 

  14. Dhara, A., Luc, D.T., Tinh, P.N.: On second-Order conditions for nonsmooth problems with constraints. Vietnam J. Math. 40(2 &3), 201–229 (2012)

    MathSciNet  Google Scholar 

  15. Gong, X.H.: Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal. 73, 3598–3612 (2010)

    Article  MathSciNet  Google Scholar 

  16. Gong, X.H.: Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342, 1455–1466 (2008)

    Article  MathSciNet  Google Scholar 

  17. Ginchev, I., Ivanov, V.I.: Second-order optimality conditions for problems with \(C^1\) data. J. Math. Anal. Appl. 340, 646–657 (2008)

    Article  MathSciNet  Google Scholar 

  18. Ivanov, V.I.: Second-order optimality conditions for vector problems with continuously Fréchet differentiable data and second-order constraint qualifications. J. Optim. Theory Appl. 166, 777–790 (2015)

    Article  MathSciNet  Google Scholar 

  19. Jiménez, B., Novo, V.: First order optimality conditions in vector optimization involving stable functions. Optimization 57(3), 449–471 (2008)

    Article  MathSciNet  Google Scholar 

  20. Jiménez, B., Novo, V.: First- and second-order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl. 284, 496–510 (2003)

    Article  MathSciNet  Google Scholar 

  21. Jeyakumar, V., Luc, D.T.: Nonsmooth vector functions and continuous optimization. Springer (2008)

  22. Jourani, A., Thibault, L.: Approximations and metric regularity in mathematical programming in Banach spaces. Math. Oper. Res. 18, 390–401 (1993)

    Article  MathSciNet  Google Scholar 

  23. Khanh, P.Q., Tuan, N.D.: Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133, 341–357 (2007)

    Article  MathSciNet  Google Scholar 

  24. Khanh, P.Q., Tuan, N.D.: First and second-order optimality conditions using approximations for nonsmooth vector optimization in Banach spaces. J. Optim. Theory Appl. 130(2), 289–308 (2006)

    Article  MathSciNet  Google Scholar 

  25. Khanh, P.Q., Tuan, N.D.: Optimality conditions using approximations for nonsmooth vector optimization problems under general inequality constraints. J. Convex Anal. 16, 169–186 (2009)

    MathSciNet  Google Scholar 

  26. Li, S.J., Meng, K.W., Penot, J.-P.: Calculus rules for derivatives of multimaps. Set-Valued Anal. 17, 21–39 (2009)

    Article  MathSciNet  Google Scholar 

  27. Luc, D.T.: Theory of vector optimization, Lect. notes in Eco. and Math. systems, Springer, Berlin 319 (1989)

  28. Luu, D.V.: Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems. J. Glob. Optim. 70, 437–453 (2018)

    Article  MathSciNet  Google Scholar 

  29. Penot, J.-P.: Second order conditions for optimization problems with constraints. SIAM J. Control. Optim. 37, 303–318 (1998)

    Article  MathSciNet  Google Scholar 

  30. Páles, Z., Zeidan, V.M.: Nonsmooth optimum problems with constraints. SIAM J. Control. Optim. 32(5), 1476–1502 (1994)

    Article  MathSciNet  Google Scholar 

  31. Rodríguez-Marín, L., Sama, M.: Scalar Lagrange multiplier rules for set-valued problems in infinite-dimensional spaces. J. Optim. Theory Appl. 156, 683–700 (2013)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  33. Sama, M.: The role of directional compactness in the existence and computation of contingent epiderivatives. J. Math. Anal. Appl. 372, 262–272 (2010)

    Article  MathSciNet  Google Scholar 

  34. Su, T.V., Hang, D.D.: Optimality conditions in terms of contingent epiderivatives for strict local Pareto minima in vector optimization problems with constraints. Positivity 25, 1737–1760 (2021). https://doi.org/10.1007/s11117-021-00842-5

    Article  MathSciNet  Google Scholar 

  35. Su, T.V., Hien, N.D.: Studniarski’s derivatives and efficiency conditions for constrained vector equilibrium problems with applications. Optimization 70(1), 121–148 (2021)

    Article  MathSciNet  Google Scholar 

  36. Su, T.V., Luu, D.V.: Second-order optimality conditions for strict pareto minima and weak efficiency for nonsmooth constrained vector equilibrium problems. Numer. Funct. Anal. Optim. 15, 1732–1759 (2022). https://doi.org/10.1080/01630563.2022.2132510

    Article  MathSciNet  Google Scholar 

  37. Su, T.V., Luu, D.V.: Higher-order efficiency conditions for constrained vector equilibrium problems. Optimization. 71, 2613–2642 (2022). https://doi.org/10.1080/02331934.2021.1873987

    Article  MathSciNet  Google Scholar 

  38. Studniarski, M.: Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control. Optim. 24, 1044–1049 (1986)

    Article  MathSciNet  Google Scholar 

  39. Tinh, P.N.: Optimality conditions for nonsmooth vector problems in normed spaces. Optimization 69(6), 1151–1186 (2020)

    Article  MathSciNet  Google Scholar 

  40. Tinh, P.N.: On optimality conditions for nonsmooth vector problems in normed spaces via generalized Hadamard directional derivatives. Optimization 72(4), 1037–1068 (2023). https://doi.org/10.1080/02331934.2022.2132824

    Article  MathSciNet  Google Scholar 

  41. Ward, D.E.: Characterizations of strict local mimima and necessary conditions for weak sharp minima. J. Optim. Theory Appl. 80(3), 551–571 (1994)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks to the anonymous referee for their valuable comments and suggestions which improve the paper. Further, the author would like to thank the Editors for the help in the processing of the article.

Funding

The author has not disclosed any funding.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, The University of Danang - University of Science and Education, Da Nang, 550000, Viet Nam

    Tran Van Su

Authors
  1. Tran Van Su
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tran Van Su.

Ethics declarations

Conflict of interest

The author declare that he has no Conflict of interest

Ethical approval

This manuscript does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

The author contributed to the study conception and design. Material preparation, data collection and analysis were performed by the author. The author read and approved the final manuscript.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Su, T. Directional Compactness, Approximations and Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints. Results Math 79, 158 (2024). https://doi.org/10.1007/s00025-024-02182-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-024-02182-8

Keywords

Search

Navigation