Optimization Letters

, Volume 13, Issue 1, pp 147–161 | Cite as

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

  • S. Khoshkhabar-amiranlooEmail author
Original Paper


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.


Levitin–Polyak well-posedness Set optimization Upper semicontinuity Huasdorff convergence Painlevé–Kuratowski convergence Cone-semicontinuity Cone-quasiconvexity 



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


  1. 1.
    Tykhonov, A.N.: On the stability of the functional optimization problems. USSR Comput. Math. Math. Phys. 6, 28–33 (1966)CrossRefGoogle Scholar
  2. 2.
    Levitin, E.S., Polyak, B.T.: Convergence of minimizing sequences in conditional extremum problems. Soviet Math. Dokl. 7, 764–767 (1966)zbMATHGoogle Scholar
  3. 3.
    Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Berlin, Springer (1993)zbMATHGoogle Scholar
  4. 4.
    Lucchetti, R., Revalski, I. (eds.): Recent Developments in Well-Posed Variational Problems, vol. 331. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  5. 5.
    Jahn, J.: Vector Optimization. Theory, Applications and Extensions, 2nd edn. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlin. Anal. 71, 3769–3778 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Long, X.J., Peng, J.W.: Generalized B-well-posedness for set optimization problems. J. Optim. Theory Appl. 157, 612–623 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Crespi, G.P., Kuroiwa, D., Rocca, M.: Convexity and global well-posedness in set-optimization. Taiwan. J. Math. 18, 1897–1908 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Khoshkhabar-amiranloo, S., Khorram, E.: Pointwise well-posedness and scalarization in set optimization. Math. Meth. Oper. Res. 82, 195–210 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dhingra, M., Lalitha, C.S.: Well-setness and scalarization in set optimization. Optim. Lett. 10, 1657–1667 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khoshkhabar-amiranloo, S., Khorram, E.: Scalarization of Levitin–Polyak well-posed set optimization problems. Optimization 66, 113–127 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Han, Y., Huang, N.-J.: Well-posedness and stability of solutions for set optimization problems. Optimization 66(1), 1–17 (2016)MathSciNetGoogle Scholar
  16. 16.
    Khoshkhabar-amiranloo, S., Khorram, E.: Scalar characterizations of cone-continuous set-valued maps. Appl. Anal. 95, 2750–2765 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kuratowski, K.: Topology, vol. 1 and 2. Academic Press, New York (1968)Google Scholar
  18. 18.
    Rockafeller, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren der Mathematischen Wisenschaften, vol. 317. Springer, Berlin (1998)CrossRefGoogle Scholar
  19. 19.
    Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Math Systems, vol. 319. Springer, Berlin (1989)Google Scholar
  20. 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)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lalitha, C.S., Chatterjee, P.: Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems. J. Glob. Optim. 59, 191–205 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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).
  23. 23.
    Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  24. 24.
    Zolezzi, T.: Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91(1), 257–266 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kuratowski, C.: Topologie, vol. 1. Panstwowe Wydawnictwo Naukowa, Warszawa (1958)zbMATHGoogle Scholar
  26. 26.
    Khoshkhabar-amiranloo, S.: Stability of minimal solutions to parametric set optimization problems. Appl. Anal. (2017).
  27. 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

Personalised recommendations