Journal of Optimization Theory and Applications

, Volume 159, Issue 3, pp 635–655 | Cite as

Scalarization in Geometric and Functional Vector Optimization Revisited



The aim of this paper is to provide a survey of some recent results in the field of optimality conditions in vector optimization with geometric and inequality/equality constraints. Moreover, the discussion we initiate leads us to consider new situations which were not previously studied.


Scalarization Pareto efficiency Michel–Penot subdifferential Approximate minima 



The work of M. Durea and R. Strugariu was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0084.


  1. 1.
    Dutta, J., Tammer, C.: Lagrangian conditions for vector optimization in Banach spaces. Math. Methods Oper. Res. 64, 521–541 (2006) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003) MATHGoogle Scholar
  5. 5.
    Zheng, X.Y., Ng, K.F.: The Fermat rule for multifunctions on Banach spaces. Math. Program., Ser. A 104, 69–90 (2005) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Durea, M., Strugariu, R.: On some Fermat rules for set-valued optimization problems. Optimization 60, 575–591 (2011) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Durea, M., Dutta, J.: Lagrange multipliers for Pareto minima in general Banach spaces. Pac. J. Optim. 4, 447–463 (2008) MathSciNetMATHGoogle Scholar
  9. 9.
    Bao, T.Q., Tammer, C.: Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications. Nonlinear Anal. 75, 1089–1103 (2012) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers for ε-Pareto solutions in vector optimization with non solid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jules, F.: Sur la somme de sous-différentiels de fonctions semi-continues inférieurement. Dissert. Math. 423 (2003) Google Scholar
  13. 13.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006) CrossRefGoogle Scholar
  14. 14.
    Durea, M.: On the existence and stability of approximate solutions of perturbed vector equilibrium problems. J. Math. Anal. Appl. 333, 1165–1176 (2007) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zălinescu, C.: Stability for a class of nonlinear optimization problems and applications. In: Nonsmooth Optimization and Related Topics, Erice, 1988, pp. 437–458. Plenum, New York (1989) CrossRefGoogle Scholar
  16. 16.
    Maeda, T.: Constraint qualifications in multiobjeetive optimization problems: differentiable case. J. Optim. Theory Appl. 80, 483–500 (1994) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999) MATHGoogle Scholar
  18. 18.
    Jiménez, B., Novo, V.: Alternative theorems and necessary optimality conditions for directionally differentiable multiobjective programs. J. Convex Anal. 9, 97–116 (2002) MathSciNetMATHGoogle Scholar
  19. 19.
    Bigi, G.: Optimality and Lagrangian regularity in vector optimization. PhD Thesis, Università di Pisa (2003) Google Scholar
  20. 20.
    Durea, M., Dutta, J., Tammer, C.: Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension. J. Math. Anal. Appl. 348, 589–606 (2008) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Huang, N.J., Li, J., Wu, S.Y.: Optimality conditions for vector optimization problems. J. Optim. Theory Appl. 142, 323–342 (2009) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969) MATHGoogle Scholar
  23. 23.
    Ye, J.J.: Nondifferentiable multiplier rules for optimization and bilevel optimization problems. SIAM J. Optim. 15, 252–274 (2004) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zheng, X.Y., Ng, K.F.: The Lagrange multiplier rule for multifunctions on Banach spaces. SIAM J. Optim. 17, 1154–1175 (2006) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Bellaassali, S., Jourani, A.: Lagrange multipliers for multiobjective programs with a general preference. Set-Valued Anal. 16, 229–243 (2008) MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Michel, P., Penot, J.-P.: Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes. C. R. Acad. Sci. Paris Sér. I Math. 12, 269–272 (1984) MathSciNetGoogle Scholar
  27. 27.
    Michel, P., Penot, J.-P.: A generalized derivative for calm and stable functions. Differ. Integral Equ. 5, 433–454 (1992) MathSciNetMATHGoogle Scholar
  28. 28.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000) MATHGoogle Scholar
  29. 29.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005) MATHGoogle Scholar
  31. 31.
    Fabian, M., Mordukhovich, B.S.: Sequential normal compactness versus topological normal compactness in variational analysis. Nonlinear Anal. 54, 1057–1067 (2003) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Marius Durea
    • 1
  • Radu Strugariu
    • 2
  • Christiane Tammer
    • 3
  1. 1.Faculty of Mathematics“Al.I. Cuza” UniversityIaşiRomania
  2. 2.Department of Mathematics and Informatics“Gh. Asachi” Technical UniversityIaşiRomania
  3. 3.Institute of MathematicsMartin-Luther-Universität Halle-WittenbergHalle (Saale)Germany

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