Acta Mathematica Vietnamica

, Volume 41, Issue 4, pp 677–694 | Cite as

Some Characterizations of Solution Sets of Vector Optimization Problems with Generalized Order

  • Nguyen Van Tuyen


We establish some necessary and sufficient conditions for optimal solutions to vector optimization problems, where the optimality notion is understood in the sense of generalized order from Mordukhovich (2006, Definition 5.53). Moreover, some criteria for the closedness and the connectedness of the set of generalized order solutions are also given. Many examples are provided to illustrate the obtained results.


Vector optimization Generalized order Necessary and sufficient optimality condition Closedness Connectedness 

Mathematics Subject Classification (2010)

90C46 90C29 90C31 


  1. 1.
    Bao, T.Q.: Subdifferential necessary conditions in set-valued optimization problems with equilibrium constraints. Optimization 63, 181–205 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bednarczuk, E.M.: Berge-type theorems for vector optimization problems. Optimization 32, 373–384 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bednarczuk, E.M.: Some stability results for vector optimization problems in partially ordered topological vector spaces, vol. III, pp 2371–2382. Tampa, Florida (1996)Google Scholar
  4. 4.
    Bednarczuk, E.M.: A note on lower semicontinuity of minimal points. Nonlinear Anal. 50, 285–297 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bednarczuk, E.M.: Upper Hölder continuity of minimal points. J. Convex Anal. 9, 327–338 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bednarczuk, E.M.: Continuity of minimal points with applications to parametric multiple objective optimization. European J. Oper. Res. 157, 59–67 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bednarczuk, E.M.: Stability analysis for parametric vector optimization problems. Diss. Math. 442, 1–126 (2007)Google Scholar
  8. 8.
    Bishop, E., Phelps, R.R.: The support functionals of a convex set. Proceedings of the Symposium in Pure Mathematics. Convexity Am. Math. Soc. 7, 27–35 (1963)MathSciNetGoogle Scholar
  9. 9.
    Henig, M.I.: The domination property in multicriteria optimization. J. Math. Anal. Appl. 114, 7–16 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huy, N.Q., Kim, D.S., Tuyen, N.V.: Existence theorems in vector optimization with generalized order. SubmittedGoogle Scholar
  11. 11.
    Luc, D.T.: Structure of the efficient point set. Proc. Am. Math. Soc. 95, 433–440 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Luc, D.T.: Theory of vector optimization. Lecture Notes in Econom. and Math. Systems 319. Springer, Berlin (1989)Google Scholar
  13. 13.
    Luc, D.T.: Contractibility of efficient point sets in normed spaces. Nonlinear Anal. 15, 527–535 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Makarov, E.K., Rachkovski, N.N.: Efficient sets of convex compacta are arcwise connected. J. Optim. Theory Appl. 110, 159–172 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. I: Basic Theory. Springer, Berlin (2006)Google Scholar
  16. 16.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. II: Applications. Springer, Berlin (2006)Google Scholar
  17. 17.
    Penot, J.P., Sterna-Karwat, A.: Parametrized multicriteria optimization: continuity and closedness of optimal multifunction. J. Math. Anal. Appl. 120, 150–168 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Mathematics in Science and Engineering, 176. Academic Press, Inc., Orlando (1985)zbMATHGoogle Scholar
  19. 19.
    Song, W.: A note on connectivity of efficient point sets. Arch. Math. 65, 540–545 (1995)CrossRefzbMATHGoogle Scholar
  20. 20.
    Song, W.: Connectivity of efficient solution sets in vector optimization of set-valued mappings. Optimization 39, 1–11 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tuyen, N.V., Yen, N.D.: On the concept of generalized order optimality. Nonlinear Anal. 75, 1592–1601 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–337 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of MathematicsHanoi Pedagogical University No. 2Phuc YenVietnam

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