Skip to main content
SpringerLink
Log in

Characterizations of generalized Levitin–Polyak well-posed set optimization problems

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we introduce the notion of generalized Levitin–Polyak (in short gLP) well-posedness for set optimization problems. We provide some characterizations of gLP well-posedness in terms of the upper Hausdorff convergence and Painlevé–Kuratowski convergence of a sequence of sets of approximate solutions, and in terms of the upper semicontinuity and closedness of an approximate solution map. We obtain some equivalence relationships between the gLP well-posedness of a set optimization problem and the gLP well-posedness of two corresponding scalar optimization problems. Also, we give some other characterizations of gLP well-posedness by two extended forcing functions and the Kuratowski noncompactness measure of the set of approximate solutions. Finally we show that certain cone-semicontinuous and cone-quasiconvex set optimization problems are gLP well-posed.

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

    Article  Google Scholar 

  2. Levitin, E.S., Polyak, B.T.: Convergence of minimizing sequences in conditional extremum problems. Soviet Math. Dokl. 7, 764–767 (1966)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Lucchetti, R., Revalski, I. (eds.): Recent Developments in Well-Posed Variational Problems, vol. 331. Kluwer Academic Publishers, Dordrecht (1995)

  5. Jahn, J.: Vector Optimization. Theory, Applications and Extensions, 2nd edn. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlin. Anal. 71, 3769–3778 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Crespi, G.P., Kuroiwa, D., Rocca, M.: Convexity and global well-posedness in set-optimization. Taiwan. J. Math. 18, 1897–1908 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Khoshkhabar-amiranloo, S., Khorram, E.: Pointwise well-posedness and scalarization in set optimization. Math. Meth. Oper. Res. 82, 195–210 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Long, X.J., Peng, J.W., Peng, Z.Y.: Scalarization and pointwise well-posedness for set optimization problems. J. Glob. Optim. 62, 763–773 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dhingra, M., Lalitha, C.S.: Well-setness and scalarization in set optimization. Optim. Lett. 10, 1657–1667 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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 

  14. Khoshkhabar-amiranloo, S., Khorram, E.: Scalarization of Levitin–Polyak well-posed set optimization problems. Optimization 66, 113–127 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Han, Y., Huang, N.-J.: Well-posedness and stability of solutions for set optimization problems. Optimization 66(1), 1–17 (2016)

    MathSciNet  Google Scholar 

  16. Khoshkhabar-amiranloo, S., Khorram, E.: Scalar characterizations of cone-continuous set-valued maps. Appl. Anal. 95, 2750–2765 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  18. Rockafeller, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren der Mathematischen Wisenschaften, vol. 317. Springer, Berlin (1998)

    Book  Google Scholar 

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

    Google Scholar 

  20. Li, Y.X.: Topological structure of efficient set of optimization problem of set-valued mapping. Chin. Ann. Math. Ser. B 15, 115–122 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Lalitha, C.S., Chatterjee, P.: Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems. J. Glob. Optim. 59, 191–205 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Crespi, G.P., Dhingra, M., Lalitha, C.S.: Pointwise and global well-posedness in set optimization: a direct approach. Ann. Oper. Res. 1–18 (2017). https://doi.org/10.1007/s10479-017-2709-7

  23. Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  24. Zolezzi, T.: Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91(1), 257–266 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kuratowski, C.: Topologie, vol. 1. Panstwowe Wydawnictwo Naukowa, Warszawa (1958)

    MATH  Google Scholar 

  26. Khoshkhabar-amiranloo, S.: Stability of minimal solutions to parametric set optimization problems. Appl. Anal. (2017). https://doi.org/10.1080/00036811.2017.1376320

  27. Crespi, G.P., Kuroiwa, D., Rocca, M.: Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization. Ann. Oper. Res. 251, 89–104 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to the anonymous referees for their valuable comments on the first version of the paper.

Author information

Authors and Affiliations

  1. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    S. Khoshkhabar-amiranloo

Authors
  1. S. Khoshkhabar-amiranloo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Khoshkhabar-amiranloo.

Additional information

This research was in part supported by a Grant from IPM (No. 95800048).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khoshkhabar-amiranloo, S. Characterizations of generalized Levitin–Polyak well-posed set optimization problems. Optim Lett 13, 147–161 (2019). https://doi.org/10.1007/s11590-018-1258-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-018-1258-6

Keywords

Search

Navigation