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

Scalarization and pointwise well-posedness for set optimization problems

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

Abstract

In this paper, we consider three kinds of pointwise well-posedness for set optimization problems. We establish some relations among the three kinds of pointwise well-posedness. By virtue of a generalized nonlinear scalarization function, we obtain the equivalence relations between the three kinds of pointwise well-posedness for set optimization problems and the well-posedness of three kinds of scalar optimization problems, respectively.

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. Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin (2005)

    Google Scholar 

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

    Google Scholar 

  3. Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)

    Book  Google Scholar 

  4. Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematics Systems, vol. 319. Springer, New York (1989)

    Google Scholar 

  5. Kuroiwa, D.: Some duality theorems of set-valued optimization with natural critera. In: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, pp. 221–228. World Scientific, River Edge (1999)

  6. Kuroiwa, D.: On set-valued optimization. Nonlinear Anal. 47, 1935–1400 (2001)

    Article  MathSciNet  Google Scholar 

  7. Kuroiwa, D.: Existence theorems of set optimization with set-valued maps. J. Inf. Optim. Sci. 24, 73–84 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Ha, T.X.D.: Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124, 187–206 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with setvaled maps. J. Math. Anal. Appl. 325, 1–18 (2007)

    Article  MathSciNet  Google Scholar 

  10. Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67, 1276–1736 (2007)

    Article  Google Scholar 

  11. Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)

    Article  MathSciNet  Google Scholar 

  12. Bednarczuck, E.M.: Well posedness of vector optimization problem. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimiation Problems. Lecture Notes in Economics and Mathematical Systems, vol. 294, pp. 51–61. Springer, Berlin (1987)

    Chapter  Google Scholar 

  13. Bednarczuck, E.M., Miglierina, E., Molho, E.: A mountain pass-type theorem for vector-valued functions. Set-Valued Var Anal. 19, 569–587 (2011)

    Article  MathSciNet  Google Scholar 

  14. Araya, Y.: Four types of nonlinear scalarizations and some applications in set optimization. Nonlinear Anal. 75, 3821–3835 (2012)

    Article  MathSciNet  Google Scholar 

  15. Tykhonov, A.N.: On the stability of the functional optimization problems. USSR Comput. Math. Phys. 6, 28–33 (1966)

    Article  Google Scholar 

  16. Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)

    Google Scholar 

  17. Bednarczuck, E.M.: An approach to well-posedness in vector optimization: consequences to stability and parametric optimization. Control Cybern. 23, 107–122 (1994)

    Google Scholar 

  18. Dentcheva, D., Helbig, S.: On variational principles, level sets, well-posedness and \(\varepsilon \)-solutions in vector optimization. J. Optim. Theory Appl. 89, 325–349 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fang, Y.P., Hu, R., Huang, N.J.: Extended \(B\)-well-posedness and Property (H) for set-valued vector optimization with convexity. J. Optim. Theory Appl. 135, 445–458 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Furi, M., Vignoli, A.: About well-posed minimization problems for functionals in metric spaces. J. Optim. Theory Appl. 5, 225–229 (1970)

    Article  MATH  Google Scholar 

  21. Loridan, P.: Well-posedness in vector optimization. In: Lucchetti, R., Recalski, J. (eds.) Recent Developments in Cariational Well-posedness Probems. Mathematics and Its Applications, vol. 331, pp. 171–192. Kluwer, Dordrecht (1995)

    Google Scholar 

  22. Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization. Math. Methods Oper. Res. 58, 375–385 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Huang, X.X.: Pointwise well-posedness of perturbed vector optimization problems in a vector-valued variational principle. J. Optim. Theory Appl. 108, 671–687 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Durea, M.: Scalarization for pointwise well-posedness vectorial problems. Math. Methods Oper. Res. 66, 409–418 (2007)

    Article  MathSciNet  Google Scholar 

  26. Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71, 3768–3788 (2008)

    MathSciNet  Google Scholar 

  27. Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822–1833 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Long, X.J., Peng, J.W.: Generalized \(B\)-well-posedness for set optimization problems. J. Optim. Theory Appl. 157, 612–623 (2013)

    Article  MathSciNet  Google Scholar 

  29. Kuratowski, K.: Topology, vols. 1 and 2. Academic Press, New York (1968)

    Google Scholar 

  30. Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor and the referees for their valuable comments and suggestions, which have improved the presentation of the paper.

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People’s Republic of China

    Xian-Jun Long

  2. School of Mathematics, Chongqing Normal University, Chongqing, 400047, People’s Republic of China

    Jian-Wen Peng

  3. College of Science, Chongqing JiaoTong University, Chongqing, 400074, People’s Republic of China

    Zai-Yun Peng

Authors
  1. Xian-Jun Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian-Wen Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zai-Yun Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian-Wen Peng.

Additional information

This research was partially supported by the National Natural Science Foundation of China (11471059, 11171363, 11301571 and 11431004), the Basic and Advanced Research Project of Chongqing (cstc2014jcyjA00037), the Education Committee Project Research Foundation of Chongqing (KJ1400618), the Program for Core Young Teacher of the Municipal Higher Education of Chongqing [(2014)47] , the Program for Chongqing Innovation Team in University (KJTD201313) and the Education Committee Project Foundation of Bayu Scholar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Long, XJ., Peng, JW. & Peng, ZY. Scalarization and pointwise well-posedness for set optimization problems. J Glob Optim 62, 763–773 (2015). https://doi.org/10.1007/s10898-014-0265-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-014-0265-0

Keywords