[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

On Set Containment Characterizations for Sets Described by Set-Valued Maps with Applications

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

Abstract

In this paper, dual characterizations of the containment of two sets involving convex set-valued maps are investigated. These results are expressed in terms of the epigraph of a conjugate function of infima associated with corresponding set-valued maps. As an application, we establish characterizations of weak and proper efficient solutions of set-valued optimization problems in the sense of vector criteria.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mangasarian, O.L.: Generalized Support Vector Machines, in Advances in Large Margin Classifiers, vol. 135–146. MIT Press, Cambridge (2000)

  2. Mangasarian, O.L.: Set containment characterization. J. Glob. Optim. 24(4), 473–480 (2002)

    Article  MathSciNet  Google Scholar 

  3. Jeyakumar, V.: Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J. Optim. 13(4), 947–959 (2003)

    Article  MathSciNet  Google Scholar 

  4. Goberna, M.A., Jeyakumar, V., Dinh, N.: Dual characterizations of set containments with strict convex inequalities. J. Glob. Optim. 34(1), 33–54 (2006)

    Article  MathSciNet  Google Scholar 

  5. Doagooei, A.R., Mohebi, H.: Dual characterizations of the set containments with strict cone-convex inequalities in Banach spaces. J. Glob. Optim. 43(4), 577–591 (2009)

    Article  MathSciNet  Google Scholar 

  6. Suzuki, S., Kuroiwa, D.: Set containment characterization for quasiconvex programming. J. Glob. Optim. 45(4), 551–563 (2009)

    Article  MathSciNet  Google Scholar 

  7. Suzuki, S.: Set containment characterization with strict and weak quasiconvex inequalities. J. Glob. Optim. 47(2), 273–285 (2010)

    Article  MathSciNet  Google Scholar 

  8. Suzuki, S., Kuroiwa, D.: On set containment characterization and constraint qualification for quasiconvex programming. J. Optim. Theory Appl. 149(3), 554–563 (2011)

    Article  MathSciNet  Google Scholar 

  9. Jeyakumar, V., Lee, G.M., Lee, J.H.: Sums of squares characterizations of containment of convex semialgebraic sets. Pac. J. Optim. 12(1), 29–42 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Kellner, K., Theobald, T., Trabandt, C.: Containment problems for polytopes and spectrahedra. SIAM J. Optim. 23(2), 1000–1020 (2013)

    Article  MathSciNet  Google Scholar 

  11. Jeyakumar, V., Ormerod, J., Womersley, R.S.: Knowledge-based semidefinite linear programming classifiers. Optim. Methods Softw. 21(5), 471–481 (2006)

    Article  MathSciNet  Google Scholar 

  12. Jeyakumar, V., Li, G., Suthaharan, S.: Support vector machine classifiers with uncertain knowledge sets via robust optimization. Optimization 63(7), 1099–1116 (2014)

    Article  MathSciNet  Google Scholar 

  13. Jeyakumar, V., Glover, B.M.: Characterizing global optimality for DC optimization problems under convex inequality constraints. J. Glob. Optim. 8(2), 171–187 (1996)

    Article  MathSciNet  Google Scholar 

  14. Jeyakumar, V., Li, G.: Characterizing robust set containments and solutions of uncertain linear programs without qualifications. Oper. Res. Lett. 38(3), 188–194 (2010)

    Article  MathSciNet  Google Scholar 

  15. Jeyakumar, V., Lee, G.M., Dinh, N.: Characterizations of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 174(3), 1380–1395 (2006)

    Article  MathSciNet  Google Scholar 

  16. Dinh, N., Goberna, M.A., López, M.A., Mo, T.H.: Farkas-type results for vector-valued functions with applications. J. Optim. Theory Appl. 173(2), 357–390 (2017)

    Article  MathSciNet  Google Scholar 

  17. Dinh, N., Goberna, M.A., Long, D.H., López, M.A.: New Farkas-type results for vector-valued functions: a non-abstract approach. J. Optim. Theory Appl. 182(1), 4–29 (2019)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Book  Google Scholar 

  20. Hernández, E., Rodríguez-Marín, L., Sama, M.: On solutions of set-valued optimization problems. Comput. Math. Appl. 60(5), 1401–1408 (2010)

    Article  MathSciNet  Google Scholar 

  21. Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. 30(3), 1487–1496 (1997)

    Article  MathSciNet  Google Scholar 

  22. Kuroiwa, D.: The natural criteria in set-valued optimization. RIMS Kokyuroku. 1031, 85–90 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Wang, F., Liu, S., Chai, Y.: Robust counterparts and robust efficient solutions in vector optimization under uncertainty. Oper. Res. Lett. 43(3), 293–298 (2015)

    Article  MathSciNet  Google Scholar 

  24. 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(3), 1311–1332 (2009)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  26. Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14(2), 534–547 (2003)

    Article  MathSciNet  Google Scholar 

  27. Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46(2), 193–211 (1997)

    Article  MathSciNet  Google Scholar 

  28. Kuroiwa, D., Popovici, N., Rocca, M.: A characterization of cone-convexity for set-valued functions by cone-quasiconvexity. Set-Valued Var Anal. 23(2), 295–304 (2015)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  30. Khoshkhabar-amiranloo, S., Khorram, E.: Scalar characterizations of cone-continuous set-valued maps. Appl. Anal. 95(12), 2750–2765 (2016)

    Article  MathSciNet  Google Scholar 

  31. 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(3), 611–626 (2013)

    Article  MathSciNet  Google Scholar 

  32. Lee, G.M., Tuan, L.A.: On \(\epsilon \)-optimality conditions for convex set-valued optimization problems. Taiwan. J. Math. 13(6A), 1787–1810 (2009)

    Article  MathSciNet  Google Scholar 

  33. Köbis, E.: On robust optimization: relations between scalar robust optimization and unconstrained multicriteria optimization. J. Optim. Theory Appl. 167(3), 969–984 (2015)

    Article  MathSciNet  Google Scholar 

  34. Sion, M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are extremely thankful to the anonymous referees and the editor for providing several suggestions to improve the paper to its current form greatly. This research was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0026/2555), the Thailand Research Fund, Grant No. RSA6080077, and Naresuan University. The third author was supported by the National Research Foundation of Korea (NRF) Grant funded by Korea government (MSIT) (NRF-2017R1E1A1A03069931).

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. Department of Applied Mathematics, Pukyong National University, Busan, 48513, Korea

    Gue Myung Lee

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. Gue Myung Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rabian Wangkeeree.

Additional information

Regina S. Burachik.

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. & Lee, G.M. On Set Containment Characterizations for Sets Described by Set-Valued Maps with Applications. J Optim Theory Appl 184, 824–841 (2020). https://doi.org/10.1007/s10957-019-01605-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01605-9

Keywords

Mathematics Subject Classification

Search

Navigation