Directional Pareto Efficiency: Concepts and Optimality Conditions

  • Teodor Chelmuş
  • Marius DureaEmail author
  • Elena-Andreea Florea


We introduce and study a notion of directional Pareto minimality with respect to a set that generalizes the classical concept of Pareto efficiency. Then, we give separate necessary and sufficient conditions for the newly introduced efficiency and several situations, concerning the objective mapping and the constraints, are considered. In order to investigate different cases, we adapt some well-known constructions of generalized differentiation; the connections with some recent directional regularities come naturally into play. As a consequence, several techniques from the study of genuine Pareto minima are considered in our specific situation.


Directional Pareto minimality Optimality conditions Directional tangent cones Directional regularity Set-valued optimization 

Mathematics Subject Classification

54C60 46G05 90C46 



This research was supported by a Grant of Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, Project No. PN-III-P4-ID-PCE-2016-0188, within PNCDI III. The authors thank two anonymous referees and the Editor-in-Chief for several suggestions that improved the presentation of the paper.


  1. 1.
    Nam, N.M., Mordukhovich, B.S.: An Easy Path to Convex Analysis and Applications. Morgan & Claypool, San Rafael (2013)zbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alzorba, S., Günther, C., Popovici, N., Tammer, C.: A new algorithm for solving planar multiobjective location problems involving the Manhattan norm. Eur. J. Oper. Res. 258, 35–46 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set Valued Var. Anal. 21, 151–176 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Huynh, V.N., Théra, M.: Directional metric regularity of multifunctions. Math. Oper. Res. 40, 969–991 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Penot, J.-P.: Analysis. From Concepts to Applications. Springer, Berlin (2016)zbMATHGoogle Scholar
  7. 7.
    Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)zbMATHGoogle Scholar
  8. 8.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Basel (1990)zbMATHGoogle Scholar
  9. 9.
    Durea, M.: First and second order optimality conditions for set-valued optimization problems. Rend. Circolo Mat. Palermo 53, 451–468 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Durea, M., Panţiruc, M., Strugariu, R.: A new type of directional regularity for mappings and applications to optimization. SIAM J. Optim. 27, 1204–1229 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Durea, M.: Estimations of the Lagrange multipliers’ norms in set-valued optimization. Pac. J. Optim. 2, 487–501 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications, Springer, Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), Vols. 330–331, Berlin (2006)Google Scholar
  16. 16.
    Penot, J.-P.: Cooperative behavior of functions, relations and sets. Math. Methods Oper. Res. 48, 229–246 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, S., Penot, J.-P., Xue, X.: Codifferential calculus. Set Valued Var. Anal. 19, 505–536 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kruger, A.Y.: About intrinsic transversality of pairs of sets. Set Valued Var. Anal. 26, 111–142 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Durea, M., Strugariu, R.: On some Fermat rules for set-valued optimization problems. Optimization 60, 575–591 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ye, J.J.: The exact penalty principle. Nonlinear Anal. Theory Methods Appl. 75, 1642–1654 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Apetrii, M., Durea, M., Strugariu, R.: A new penalization tool in scalar and vector optimizations. Nonlinear Anal. Theory, Methods Appl. 107, 22–33 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Cibulka, R., Durea, M., Panţiruc, M., Strugariu, R.: On the stability of the directional regularity. Set Valued Var. Anal. (2019). Google Scholar
  23. 23.
    Kuroiwa, D.: On set-valued optimization. In: Proceedings of the 3rd World Congress of Nonlinear Analysts 47, Part 2 (Catania, 2000), pp. 1395–1400 (2001)Google Scholar
  24. 24.
    Kobis, E., Le, T.T., Tammer, C.: A generalized scalarization method in set optimization with respect to variable domination structures. Vietnam J. Math. 46, 95–125 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zaslavski, A.J.: Numerical Optimization with Computational Errors. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
  27. 27.
    Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems. Math. Program. 41, 73–96 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics“Al. I. Cuza” UniversityIasiRomania
  2. 2.“Octav Mayer” Institute of Mathematics of the Romanian AcademyIasiRomania

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