Optimization Letters

, Volume 8, Issue 7, pp 2021–2037 | Cite as

Scalarization of constraints system in some vector optimization problems and applications

Original Paper

Abstract

In this work we employ a new method to penalize a constrained non solid vector optimization problem by means of a scalarization functional applied to the constraints system. Then, we formulate optimality conditions which mainly use several types of regularity for single and set-valued maps. In order to motivate our demarche, we discuss in detail the assumptions used in the main results and we show how it can be verified.

Keywords

Penalization Scalarization of constraints Pareto efficiency  Bouligand cone 

References

  1. 1.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkäuser, Basel (1990)MATHGoogle Scholar
  2. 2.
    Bao, T.Q., Mordukhovich, B.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Mathe. Program. 122, 101–138 (2010)MathSciNetGoogle Scholar
  3. 3.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  4. 4.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  5. 5.
    Durea, M.: Estimations of the Lagrange multipliers’ norms in set-valued optimization. Pac. J. Optim. 2, 487–501 (2006)MathSciNetMATHGoogle Scholar
  6. 6.
    Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)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., Strugariu, R.: Necessary optimality conditions for weak sharp minima in set-valued optimization. Nonlinear Anal. Theory Methods Appl. 73, 2148–2157 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Durea, M., Strugariu, R.: On parametric vector optimization via metric regularity of constraint systems. Math. Methods Oper. Res. 74, 409–425 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    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)Google Scholar
  12. 12.
    Gerstewitz, C., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme, Wiss. Z. Tech. Hochsch. Ilmenau, 31, 61–81 (1985)Google Scholar
  13. 13.
    Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)MATHGoogle Scholar
  14. 14.
    Hernández, E., Rodríguez-Marín, L., Sama, M.: Scalar multipliers rules in set-valued optimization. Comput. Math. Appl. 57, 1286–1293 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hernández, E., Rodríguez-Marín, L., Khan, A.A., Sama, M.: Computation formulas and nultiplier rules for graphical derivatives in separable Banach spaces. Nonlinear Anal. Theory Methods Appl. 71, 4241–4250 (2009)CrossRefMATHGoogle Scholar
  16. 16.
    Ioffe, A.D.: Optimality alternative: a non-variational approach to necessary conditions. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications Nonconvex Optimization and Its Applications, vol. 79, pp. 531–552 (2005)Google Scholar
  17. 17.
    Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2004)MATHGoogle Scholar
  18. 18.
    Khan, A.K., Sama, M.: A multiplier rule for stable problems in vector optimization. J. Convex Anal. 19, 525–539 (2012)MathSciNetMATHGoogle Scholar
  19. 19.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation, vol. I: basic theory, vol. 330. Springer, Berlin (2006) [Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics)]Google Scholar
  20. 20.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation, vol. II: applications, vol. 331. Springer, Berlin (2006) [Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics)]Google Scholar
  21. 21.
    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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rodríguez-Marín, L., Sama, M.: Scalar Lagrange multiplier rules for set-valued problems in infinite-dimensional spaces. J. Optim. Theory Appl. 156, 683–700 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rockafellar, R.T.: Proto-differentiability of set-valued mappings and its applications in optimization. Ann. Inst. H. Poincaré 6, 449–482 (1989)MathSciNetMATHGoogle Scholar
  24. 24.
    Tammer, C., Zălinescu, C.: Lipschitz properties of the scalarization function and applications. Optimization 59, 305–319 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ursescu, C.: Tangency and openness of multifunctions in Banach spaces. Analele Ştiinţifice ale Universităţii “Al. I. Cuza” Iaşi 34, 221–226 (1988)MathSciNetMATHGoogle Scholar
  26. 26.
    Ye, J.J.: The exact penalty principle. Nonlinear Anal. Theory Methods Appl. 75, 1642–1654 (2012)CrossRefMATHGoogle Scholar
  27. 27.
    Zaslavski, A.J.: An approximate exact penalty in constrained vector optimization on metric spaces. J. Optim. Theory Appl. doi:10.1007/s10957-013-0288-6

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of MathematicsAl. I. Cuza UniversityIasiRomania
  2. 2.Department of Mathematics and InformaticsGh. Asachi Technical UniversityIasiRomania

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