Optimization Letters

, Volume 13, Issue 1, pp 55–68 | Cite as

Optimality conditions and minimax properties in set optimization

  • María Alonso
  • Elvira HernándezEmail author
  • Elena Pereira
Original Paper


In this paper, we consider a generalization of the Gerstewitz’s function to present several optimality conditions and existence theorems for a set optimization problem without convexity assumptions. A characterization of set solutions for a set-valued optimization problem is given via minimax inequalities.


Set-valued maps Optimality conditions Minimax Set optimization 



This research was partially supported by project ETSI Industriales (UNED) 2018-MAT11.


  1. 1.
    Chen, G.Y., Göh, C.J., Yang, X.Q.: Vector network equilibrium problems and nonlinear scalarization methods. Math. Methods Oper. Res. 49, 239–253 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chen, C.R., Li, M.H.: H\(\ddot{o}\)lder continuity of solutions to parametric vector equilibrium problems with nonlinear scalarization. Numer. Funct. Anal. Optim. 35, 685–707 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60, 1399–1419 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hernández, E.: Problemas de optimización en análisis de multifunciones. PhD. (2005)Google Scholar
  6. 6.
    Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67, 1726–1736 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jahn, J.: Vectorization in set optimization. J. Optim. Theory Appl. 167, 783–795 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Khan, A.A., Tammer, C., Zǎlinescu, C.: Set-valued Optimization: An Introduction with Applications. Springer, Heidelberg (2015)CrossRefzbMATHGoogle Scholar
  10. 10.
    K\(\ddot{\text{o}}\)bis, E., K\(\ddot{\text{ o }}\)bis, A.: Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization. Optimization 65, 1805–1827 (2016)Google Scholar
  11. 11.
    Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. 30, 1487–1496 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuwano, I.: Some minimax theorems of set-valued maps and their applications. Nonlinear Anal. 109, 85–102 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li, S.J., Yang, X.Q., Chen, G.Y.: Nonconvex vector optimization of set-valued mappings. J. Math. Anal. Appl. 283, 337–350 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)Google Scholar
  15. 15.
    Xu, Y.D., Li, S.J.: A new nonlinear scalarization function and applications. Optimization 64, 207–231 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, Y., Chen, T.: Minimax problems for set-valued mappings with set-relations. Numer. Algebra Contr. Optim. 4, 327–340 (2014)CrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Universidad Nacional de Educación a DistanciaMadridSpain

Personalised recommendations