Skip to main content
SpringerLink
Log in

Sequential characterizations of approximate solutions in convex vector optimization problems with set-valued maps

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

This paper deals with a convex vector optimization problem with set-valued maps. In the absence of constraint qualifications, it provides, by the scalarization theorem, sequential Lagrange multiplier conditions characterizing approximate weak Pareto optimal solutions for the problem in terms of the approximate subdifferentials of the marginal function associated with corresponding set-valued maps. The paper shows also that this result yields the approximate Lagrange multiplier condition for the problem under a new constraint qualification which is weaker than the Slater-type constraint qualification. Illustrative examples are also provided to discuss the significance of the sequential conditions.

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. Andreani, R., Martínez, J.M., Svaiter, B.F.: A new sequential optimality condition for constrained optimization and algorithmic consequences. SIAM J. Optim. 20, 3533–3554 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and algorithmic consequences. SIAM J. Optim. 26, 96–110 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Bai, F., Wu, Z., Zhu, D.: Sequential Lagrange multiplier condition for \(\epsilon \)-optimal solution in convex programming. Optimization 57, 669–680 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Baier, J., Jahn, J.: On subdifferentials of set-valued maps. J. Optim. Theory Appl. 100, 233–240 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Boţ, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010)

    MATH  Google Scholar 

  6. Boţ, R.I., Csetnek, E.R., Wanka, G.: Sequential optimality conditions in convex programming via perturbation approach. J. Convex Anal. 15, 149–164 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Brondsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)

    MathSciNet  MATH  Google Scholar 

  8. Burachik, R.S., Jeyakumar, V.: A simple closure condition for the normal cone intersection formula. Proc. Am. Math. Soc. 133, 1741–1748 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Durea, J., Dutta, J., Tammer, C.: Lagrange multipliers for \(\varepsilon \)-pareto solutions in vector optimization with nonsolid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Dutta, J., Deb, K., Tulshyan, R., Arora, R.: Approximate KKT points and a proximity measure for termination. J. Glob. Optim. 56, 1463–1499 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming. SIAM J. Optim. 20, 1311–1332 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Flores-Bazán, F.: Optimality conditions in non-convex set-valued optimization. Math. Methods Oper. Res. 53, 403–417 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)

    MATH  Google Scholar 

  14. Götz, A., Jahn, J.: The Lagrange multiplier rule in set-valued optimization. SIAM J. Optim. 10, 331–344 (1999)

    MathSciNet  MATH  Google Scholar 

  15. Ha, T.X.D.: Lagrange multipliers for set-valued optimization problems associated with coderivatives. J. Math. Anal. Appl. 311, 647–663 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Haeser, G., Melo, V.V.: Convergence detection for optimization algorithms: approximate-KKT stopping criterion when Lagrange multipliers are not available. Oper. Res. Lett. 43, 484–488 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Hernández, E., Jiménez, B., Novo, V.: Weak and proper efficiency in set-valued optimization on real linear spaces. J. Convex. Anal. 14, 275–296 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Hernández, E., Rodríguez-Marín, L., Sama, M.: Scalar multiplier rules in set-valued optimization. Comput. Math. Appl. 57, 1286–1293 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Huang, X.X., Yao, J.C.: Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems. J. Glob. Optim. 55, 611–626 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Heidelberg (2004)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Jeyakumar, V.: Asymptotic dual conditions characterizing optimality for convex programs. J. Optim. Theory Appl. 93, 153–165 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Jeyakumar, V., Dinh, N., Lee, G.M.: A new closed cone constraint qualification for convex Optimization. In: Applied Mathematics Research Report AMR04/8, School of Mathematics, University of New South Wales (2004)

  24. Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14, 534–547 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Jeyakumar, V., Wu, Z.Y., Lee, G.M., Dinh, N.: Liberating the subgradient optimality conditions from constraint qualifications. J. Glob. Optim. 36, 127–137 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Khoshkhabar-amiranloo, S., Khorram, E.: Scalar characterizations of cone-continuous setvalued maps. Appl. Anal. 95, 2750–2765 (2016)

    MathSciNet  MATH  Google Scholar 

  27. Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. 30, 1487–1496 (1997)

    MathSciNet  MATH  Google Scholar 

  28. Lee, G.M., Tuan, L.A.: On \(\epsilon \)-optimality conditions for convex set-valued optimization problems. Taiwan. J. Math. 13, 1787–1810 (2009)

    MathSciNet  MATH  Google Scholar 

  29. Li, T.Y., Xu, Y.H., Zhu, C.X.: \(\epsilon \)-strictly efficient solutions of vector optimization problems with set-valued maps. Asia. Pac. J. Oper. Res. 24, 841–854 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Long, X.J., Li, X.B., Zeng, J.: Lagrangian conditions for approximate solutions on nonconvex set-valued optimization problems. Optim. Lett. 7, 1847–1856 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Math Systems, vol. 319. Springer, Berlin (1989)

    Google Scholar 

  32. Martínez, J.M., Svaiter, B.F.: A practical optimality condition without constraint qualifications for nonlinear programming. J. Optim. Theory Appl. 118, 117–133 (2003)

    MathSciNet  MATH  Google Scholar 

  33. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, New York (2018)

    MATH  Google Scholar 

  34. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)

    MathSciNet  MATH  Google Scholar 

  35. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1997)

    MATH  Google Scholar 

  36. Rong, W.D., Wu, Y.N.: \(\epsilon \)-weak minimal solutions of vector optimization problems with set-valued maps. J. Optim. Theory Appl. 106, 569–579 (2000)

    MathSciNet  Google Scholar 

  37. Sisarat, N., Wangkeeree, R., Lee, G.M.: On set containment characterizations for sets described by set-valued maps with applications. J. Optim. Theory Appl. (2019). https://doi.org/10.1007/s10957-019-01605-9

    Article  MATH  Google Scholar 

  38. Taa, A.: \(\varepsilon \)-subdifferentials of set-valued maps and \(\varepsilon \)-weak Pareto optimality for multiobjective optimization. Math. Methods Oper. Res. 62, 187–209 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Thibault, L.: Sequential convex subdifferential calculus and sequential Lagrange multipliers. SIAM J. Control Optim. 35, 1434–1444 (1997)

    MathSciNet  MATH  Google Scholar 

  40. Tuan, L.A.: \(\epsilon \)-optimality conditions for vector optimization problems with set-valued maps. Numer. Func. Anal. Optim. 31, 78–95 (2010)

    MathSciNet  Google Scholar 

  41. Tuyen, N.V., Yao, J.-C., Wen, C.-F.: A note on approximate Karush–Kuhn–Tucker conditions in locally Lipschitz multiobjective optimization. Optim. Lett. 13, 163–174 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Zheng, X.Y., Ng, K.F.: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim. 17, 1154–1175 (2006)

    MathSciNet  MATH  Google Scholar 

  43. Zhou, Z.A., Yang, X.M., Peng, J.W.: \(\epsilon \)-optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces. Optim. Lett. 8, 1047–1061 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to express their sincere thanks to anonymous reviewers by their valuable corrections and comments, which greatly improved the readability of the paper. We also are grateful to Professor Gue Myung Lee for helpful discussions on the paper. The first-named author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0026/2555). The second-named author was supported by Naresuan University.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

    Nithirat Sisarat & Rabian Wangkeeree

  2. Research center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok, Thailand

    Rabian Wangkeeree

  3. Graduate School of Science and Technology, Niigata University, Niigata, 950-2181, Japan

    Tamaki Tanaka

Authors
  1. Nithirat Sisarat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rabian Wangkeeree
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tamaki Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rabian Wangkeeree.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sisarat, N., Wangkeeree, R. & Tanaka, T. Sequential characterizations of approximate solutions in convex vector optimization problems with set-valued maps. J Glob Optim 77, 273–287 (2020). https://doi.org/10.1007/s10898-019-00864-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-019-00864-0

Keywords

Mathematics Subject Classification

Search

Navigation