A barrier method in convex vector optimization with generalized inequality constraints

  • Marius DureaEmail author
  • Radu Strugariu
Original paper


In this note we present a barrier method for vector optimization problems with inequality constraints. To this aim, we firstly investigate some constraint qualification conditions and we compare them to the corresponding ones in literature. Then, we define a barrier function and observe that its basic properties do work for fairly general situations, while for meaningful convergence results of the associated barrier method we should restrict ourselves to convex case and finite dimensional setting.


Openness Vector convexity Gerstewitz scalarization Barrier method 



This work was supported by a grant of Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0188, within PNCDI III.


  1. 1.
    Bonnans, F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Durea, M.: Estimations of the Lagrange multipliers’ norms in set-valued optimization. Pac. J. Optim. 2, 487–501 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Durea, M., Strugariu, R.: Optimality conditions and a barrier method in optimization with convex geometric constraint. Optim. Lett. 12, 923–931 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dutta, J.: Barrier method in nonsmooth convex optimization without convex representation. Optim. Lett. 9, 1177–1185 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dutta, J., Lalitha, C.S.: Optimality conditions in convex optimization revisited. Optim. Lett. 7, 221–229 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)zbMATHGoogle Scholar
  7. 7.
    Lasserre, J.B.: On representation of the feasible set in convex optimization. Optim. Lett. 4, 1–5 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lasserre, J.B.: On convex optimization without convex representation. Optim. Lett. 5, 549–556 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lasserre, J.B.: Erratum to: On convex optimization without convex representation. Optim. Lett. 8, 1795–1796 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, II: Applications. A Series of Comprehensive Studies in Mathematics, vol. 330 and 331. Sringer, Berlin (2006)CrossRefGoogle Scholar
  11. 11.
    Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics“Alexandru Ioan Cuza” UniversityIasiRomania
  2. 2.“Octav Mayer” Institute of Mathematics of the Romanian AcademyIasiRomania
  3. 3.Department of Mathematics“Gheorghe Asachi” Technical UniversityIasiRomania

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