Journal of Global Optimization

, Volume 68, Issue 4, pp 899–923 | Cite as

Vectorial penalization for generalized functional constrained problems



In this paper we use a double penalization procedure in order to reduce a set-valued optimization problem with functional constraints to an unconstrained one. The penalization results are given in several cases: for weak and strong solutions, in global and local settings, and considering two kinds of epigraphical mappings of the set-valued map that defines the constraints. Then necessary and sufficient conditions are obtained separately in terms of Bouligand derivatives of the objective and constraint mappings.


Set-valued vector optimization Penalization Bouligand derivative of set-valued maps Necessary optimality conditions 

Mathematics Subject Classification

49J53 49K27 90C46 



This research was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0019.


  1. 1.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkäuser, Basel (1990)MATHGoogle Scholar
  2. 2.
    Apetrii, M., Durea, M., Strugariu, R.: On subregularity properties of set-valued mappings. Applications to solid vector optimization. Set-Valued Var. Anal. 21, 93–126 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Apetrii, M., Durea, M., Strugariu, R.: A new penalization tool in scalar and vector optimizations. Nonlinear Anal. Theory Methods Appl. 107, 22–33 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  6. 6.
    Durea, M., Nguyen, H.T., Strugariu, R.: Metric regularity of epigraphical multivalued mappings and applications to vector optimization. Math. Program. Ser. B 139, 139–159 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Durea, M., Panţiruc, M., Strugariu, R.: Minimal time function with respect to a set of directions. Basic properties and applications. Optim. Methods Softw. 31, 535–561 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Durea, M., Strugariu, R.: Openness stability and implicit multifunction theorems. Applications to variational systems. Nonlinear Anal. Theory Methods Appl. 75, 1246–1259 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Durea, M., Strugariu, R.: Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions. J. Glob. Optim. 56, 587–603 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Durea, M., Strugariu, R.: Scalarization of constraints system in some vector optimization problems and applications. Optim. Lett. 8, 2021–2037 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Durea, M., Strugariu, R.: Metric subregularity of composition set-valued mappings with applications to fixed point theory. Set-Valued Var. Anal. 24, 231–251 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Durea, M., Strugariu, R., Tammer, C.: On set-valued optimization problems with variable ordering structure. J. Glob. Optim. 61, 745–767 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Jimenez, B., Novo, V.: A notion of local proper efficiency in the Borwein sense in vector optimization. ANZIAM J. 22, 75–89 (2003)CrossRefMATHGoogle Scholar
  14. 14.
    Kuroiwa, D.: The natural criteria in set-valued optimization. RIMS Kokyuroku 1998, 85–90 (1031)MathSciNetMATHGoogle Scholar
  15. 15.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications, Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), Vol. 330 and 331. Springer, Berlin (2006)Google Scholar
  16. 16.
    Nam, N.M., Zălinescu, C.: Variational analysis of directional minimal time functions and applications to location problems. Set-Valued Var. Anal. 21, 405–430 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ngai, H.V., Nguyen, H.T., Théra, M.: Metric regularity of the sum of multifunctions and applications. J. Optim. Theory Appl. 160, 355–390 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rockafellar, R.T.: Proto-differentiability of set-valued mappings and its applications in optimization. Ann. Inst. H. Poincaré 6, 449–482 (1989)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ursescu, C.: Tangency and openness of multifunctions in Banach spaces. An. Ştiinţifice ale Univ. ”Al. I. Cuza” Iaşi 34, 221–226 (1988)MathSciNetMATHGoogle Scholar
  20. 20.
    Ye, J.J.: The exact penalty principle. Nonlinear Anal. Theory Methods Appl. 75, 1642–1654 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics“Al. I. Cuza” UniversityIasiRomania
  2. 2.Department of Mathematics“Gh. Asachi” Technical UniversityIasiRomania

Personalised recommendations