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Journal of Optimization Theory and Applications

, Volume 158, Issue 2, pp 363–384 | Cite as

Variational Sets of Perturbation Maps and Applications to Sensitivity Analysis for Constrained Vector Optimization

  • N. L. H. Anh
  • P. Q. Khanh
Article

Abstract

We consider sensitivity analysis in terms of variational sets for nonsmooth vector optimization. First, relations between variational sets, or their minima/weak minima, of a set-valued map and that of its profile map are obtained. Second, given an objective map, relationships between the above sets of this objective map and that of the perturbation map and weak perturbation map are established. Finally, applications to constrained vector optimization are given. Many examples are provided to illustrate the essentialness of the imposed assumptions and some advantages of our results.

Keywords

Nonsmooth vector optimization Sensitivity analysis Perturbation maps Weak perturbation maps Variational sets Singular variational sets 

Notes

Acknowledgements

This research was supported by the Vietnam National University Hochiminh City. A part of the work of the second author was completed during his stay as a visiting professor at the Vietnam Institute for Advanced Study in Mathematics, whose hospitality is gratefully acknowledged. The authors are indebted to an anonymous referee for many valuable detailed remarks which have helped improve the paper significantly.

References

  1. 1.
    Fiacco, A.V.: Introduction to sensitivity and stability analysis. In: Nonlinear Programming. Academic Press, New York (1983) Google Scholar
  2. 2.
    Tanino, T.: Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56, 479–499 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Tanino, T.: Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim. 26, 521–536 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Shi, D.S.: Contingent derivative of the perturbation map in multiobjective optimization. J. Optim. Theory Appl. 70, 385–396 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Shi, D.S.: Sensitivity analysis in convex vector optimization. J. Optim. Theory Appl. 77, 145–159 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in vector optimization. J. Optim. Theory Appl. 89, 713–730 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in parametrized convex vector optimization. J. Math. Anal. Appl. 202, 511–522 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Klose, J.: Sensitivity analysis using the tangent derivative. Numer. Funct. Anal. Optim. 13, 143–153 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Wang, Q.L., Li, S.J.: Second-order contingent derivative of the perturbation map in multiobjective optimization. Fixed Point Theory Appl. 2011, 857520 (2011) Google Scholar
  10. 10.
    Sun, X.K., Li, S.J.: Lower Studniarski derivative of the perturbation map in parametrized vector optimization. Optim. Lett. 5, 601–614 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990) zbMATHGoogle Scholar
  12. 12.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, 3rd edn. Springer, Berlin (2009) zbMATHGoogle Scholar
  13. 13.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Springer, Berlin (2006) Google Scholar
  14. 14.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications. Springer, Berlin (2006) Google Scholar
  15. 15.
    Göpfert, A., Tammer, C., Riahi, H., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003) zbMATHGoogle Scholar
  16. 16.
    Luc, D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99–111 (1991) zbMATHCrossRefGoogle Scholar
  17. 17.
    Li, S.J., Teo, K.L., Yang, X.Q.: Higher-order optimality conditions for set-valued optimization. J. Optim. Theory Appl. 137, 533–553 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46, 193–211 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Bednarczuk, E.M., Song, W.: Contingent epiderivative and its applications to set-valued maps. Control Cybern. 27, 375–386 (1998) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Jahn, J., Khan, A.A., Zeilinger, P.: Second-order optimality conditions in set optimization. J. Optim. Theory Appl. 125, 331–347 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Jahn, J., Khan, A.A.: Generalized contingent epiderivatives in set valued optimization: optimality conditions. Numer. Funct. Anal. Optim. 23, 807–831 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Wang, Q.L., Li, S.J.: Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency. Numer. Funct. Anal. Optim. 30, 849–869 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Crespi, G.P., Ginchev, I., Rocca, M.: First-order optimality conditions in set-valued optimization. Math. Methods Oper. Res. 63, 87–106 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Khan, A.A., Raciti, F.: A multiplier rule in set-valued optimization. Bull. Aust. Math. Soc. 68, 93–100 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kasimbeyli, R.: Radial epiderivatives and set-valued optimization. Optimization 58, 521–534 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Flores-Bazan, F.: Radial epiderivatives and asymptotic function in nonconvex vector optimization. SIAM J. Optim. 14, 284–305 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Anh, N.L.H., Khanh, P.Q., Tung, L.T.: Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization. Nonlinear Anal. TMA 74, 7365–7379 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Anh, N.L.H., Khanh, P.Q.: Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives. J. Global Optim. (2012). doi: 10.1007/s10898-012-9861-z. Online First Google Scholar
  29. 29.
    Studniarski, M.: Higher-order necessary optimality conditions in terms of Neustadt derivatives. Nonlinear Anal. 47, 363–373 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Ginchev, I.: Higher-order optimality conditions in nonsmooth optimization. Optimization 51, 47–72 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Ginchev, I.: Higher-order optimality conditions in nonsmooth vector optimization. J. Stat. Manag. Syst. 5, 321–339 (2002). In: Cambini, A., Dass, B.K., Martein, L., Generalized Convexity, Generalized Monotonicity, Optimality Conditions and Duality in Scalar and Vector Optimization MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Khanh, P.Q., Tuan, N.D.: Variational sets of multivalued mappings and a unified study of optimality conditions. J. Optim. Theory Appl. 139, 45–67 (2008) Google Scholar
  33. 33.
    Khanh, P.Q., Tuan, N.D.: Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization. J. Optim. Theory Appl. 139, 243–261 (2008) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Anh, N.L.H., Khanh, P.Q., Tung, L.T.: Variational sets: calculus and applications to nonsmooth vector optimization. Nonlinear Anal. 74, 2358–2379 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Diem, H.T.H., Khanh, P.Q., Tung, L.T.: On higher-order sensitivity analysis in nonsmooth vector optimization. J. Optim. Theory Appl., to appear Google Scholar
  36. 36.
    Anh, N.L.H., Khanh, P.Q.: Higher-order radial epiderivatives and optimality conditions in nonsmooth vector optimization. Submitted for publication Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and ComputingUniversity of Science of Hochiminh CityHochiminh CityVietnam
  2. 2.Department of MathematicsInternational University of Hochiminh CityHochiminh CityVietnam

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