Advertisement

Positivity

, Volume 20, Issue 1, pp 41–60 | Cite as

Higher-order optimality conditions for set-valued optimization with ordering cones having empty interior using variational sets

  • Nguyen Le Hoang AnhEmail author
Article

Abstract

In this paper, we first establish chain rules and sum rules for variational sets of type 2. For their applications, optimality conditions of two particular optimization problems are discussed. Then, we obtain higher-order optimality conditions for proper Henig solutions of a set-valued optimization problem in terms of variational sets of type 2 when ordering cones have empty interior.

Keywords

Set-valued optimization Higher-order optimality conditions Calculus rules Proper Henig solutions Variational sets Nearly subconvexlikeness 

Mathematics Subject Classification

32F17 46G05 54C60 90C46 

Notes

Acknowledgments

This study was supported by the project of the Moravian-Silesian Region, Czech Republic Reg. No. 02692/2014/RRC. The author is grateful to an anonymous referee for his valuable comments which helped to improve the previous manuscript.

References

  1. 1.
    Anh, N.L.H., Khanh, P.Q.: Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives. J. Global Optim. 56, 519–536 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Anh, N.L.H., Khanh, P.Q.: Variational sets of perturbation maps and applications to sensitivity analysis for constrained vector optimization. J. Optim. Theory Appl. 158, 363–384 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Anh, N.L.H., Khanh, P.Q., Tung, L.T.: Variational sets: calculus rules and applications to nonsmooth vector optimization. Nonlinear Anal. TMA 74, 2358–2379 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Aubin, J.P., Frankowska, H.: Set-valued analysis. Birkhauser, Boston (1990)zbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I : quasi relative interiors and duality theory. Math. Program. 57, 15–48 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Borwein, J.M., Zhuang, D.: Super-efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Bot, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi-relative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Cammaroto, F., Di Bella, B.: Separation theorem based on the quasi relative interior and application to duality theory. J. Optim. Theory Appl. 125, 223–229 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Chen, C.R., Li, S.J., Teo, K.L.: Higher-order weak epiderivatives and applications to duality and optimality conditions. Comp. Math. Appl. 57, 1389–1399 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    De Araujo, A.P., Monteiro, P.K.: On programming when the positive cone has an empty interior. J. Optim. Theory Appl. 67, 395–410 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Diem, H.T.H., Khanh, P.Q., Tung, L.T.: On higher-order sensitivity analysis in nonsmooth vector optimization. J. Optim. Theory Appl. 162, 463–488 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Grad, A.: Generalized duality and optimality conditions. Editura Mega, Cluj-Napoca (2010)Google Scholar
  15. 15.
    Jahn, J.: Vector optimization: theory. Applications and extensions. Springer, New York (2004)CrossRefGoogle Scholar
  16. 16.
    Jahn, J., Khan, A.A.: Generalized contingent epiderivatives in set valued optimization : optimality conditions. Numer. Funct. Anal. Optim. 23, 807–831 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Jahn, J., Khan, A.A.: Some calculus rules for contingent epiderivatives. Optimization 50, 113–125 (2003)CrossRefMathSciNetGoogle Scholar
  18. 18.
    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
  19. 19.
    Khanh, P.Q., Tuan, N.D.: Higher-order variational sets and higher-order optimality conditions for proper efciency in set-valued nonsmooth vector optimization. J. Optim. Theory Appl. 139, 243–261 (2008)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Li, S.J., Chen, C.R.: Higher-order optimality conditions for Henig efficient solutions in set-valued optimization. J. Math. Anal. Appl. 323, 1184–1200 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Li, S.J., Meng, K.W., Penot, J.-P.: Calculus rules for derivatives of multimaps. Set-Valued Anal. 17, 21–39 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Luc, D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99–111 (1991)CrossRefzbMATHGoogle Scholar
  24. 24.
    Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation. Basic theory, I edn. Springer, Berlin (2006)Google Scholar
  26. 26.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation. Applications, II edn. Springer, Berlin (2006)Google Scholar
  27. 27.
    Rockafellar, R.T., Wets, R.J.B.: Variational analysis, 3rd edn. Springer, Berlin (2009)zbMATHGoogle Scholar
  28. 28.
    Taa, A.: Set-valued derivatives of multifunctions and optimality conditions. Numer. Func. Anal. Optimi. 19, 121–140 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Wang, Q.L., Li, S.J.: Higher-order weakly generalized adjacent epiderivatives and applications to duality of set-valued optimization. J. Inequal. Appl. 462637 (2009)Google Scholar
  30. 30.
    Wang, Q.L., Li, S.J.: Generailized higher-order optimality conditions for set-valued optimization under Henig efficiency. Numer. Funct. Anal. Optim. 30, 849–869 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Wang, Q.L., Li, S.J., Teo, K.L.: Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization. Optim. Lett. 4, 425–437 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OstravaOstravaCzech Republic
  2. 2.Department of Optimization and System TheoryUniversity of Science, Vietnam National University Hochiminh CityHo Chi Minh CityVietnam

Personalised recommendations