Journal of Global Optimization

, Volume 68, Issue 3, pp 587–600 | Cite as

Coradiant sets and \(\varepsilon \)-efficiency in multiobjective optimization

  • Abbas Sayadi-bander
  • Latif Pourkarimi
  • Refail Kasimbeyli
  • Hadi Basirzadeh


This paper studies \(\varepsilon \)-efficiency in multiobjective optimization by using the so-called coradiant sets. Motivated by the nonlinear separation property for cones, a similar separation property for coradiant sets is investigated. A new notion, called Bishop–Phelps coradiant set is introduced and some appropriate properties of this set are studied. This paper also introduces the notions of \(\varepsilon \)-dual and augmented \(\varepsilon \)-dual for Bishop and Phelps coradiant sets. Using these notions, some scalarization and characterization properties for \(\varepsilon \)-efficient and proper \(\varepsilon \)-efficient points are proposed.


Multiobjective optimization Efficiency \(\varepsilon \)-Efficiency Bishop and Phelps coradiant set Scalarization 


  1. 1.
    Bishop, E., Phelps, R.R.: The support functionals of a convex set. Proc. Sympos. Pure Math. 7, 27–35 (1962)CrossRefMATHGoogle Scholar
  2. 2.
    Dentcheva, D., Helbig, S.: On variational principles, level sets, well-posedness, and \(\varepsilon -\)solutions in vector optimization. J. Optim. Theory Appl. 89(2), 325–349 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  4. 4.
    Eichfelder, G.: Adaptive Scalarization Methods in Moltiobjective Optimization. Springer-verlag, Berlin (2008)CrossRefMATHGoogle Scholar
  5. 5.
    Flores-Bazan, F., Hernandez, E.: Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. J. Global Optim. 56(2), 299–315 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gutierrez, C., Jimenez, B., Novo, V.: Multiplier rules and saddle-point theorems for Helbigs approximate solutions in convex Pareto problems. J. Global Optim. 32(3), 367–383 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gutierrez, C., Jimenez, B., Novo, V.: On approximate efficiency in multiobjective programming. Math. Methods Oper. Res. 64, 165–185 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gutierrez, C., Jimenez, B., Novo, V.: A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17, 688–710 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Helbig, S.: On a New Concept for \(\varepsilon \)-Efficiency: talk at “Optimization Days 1992”, Montreal (1992)Google Scholar
  10. 10.
    Idrissi, H., Loridan, P., Michelot, C.: Approximation of solutions for location problems. J. Optim. Theory Appl. 56(1), 127–143 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Isac, G.: The Ekelands principle and the Pareto \(\varepsilon -\)efficiency. Lect. Notes Econom. Math. Syst. 432, 148–163 (1996)CrossRefMATHGoogle Scholar
  12. 12.
    Kasimbeyli, R.: A conic scalarization method in multi-objective optimization. J. Global Optim. 56(2), 279–297 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kasimbeyli, R.: A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J. Optim. 20, 1591–1619 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kasimbeyli, R., Kasimbeyli, N.: The nonlinear separation theorem and a representation theorem for Bishop Phelps Cones. In: Modelling, Computation and Optimization in Information Systems and Management Sciences. Springer International Publishing. 419–430 (2015)Google Scholar
  15. 15.
    Kutateladze, S.S.: Convex \(\varepsilon -\)programming. Soviet Math. Dokl. 20(2), 391–393 (1979)MATHGoogle Scholar
  16. 16.
    Liu, J.C.: \(\varepsilon \)-duality theorem of nondifferentiable nonconvex multiobjective programming. J. Optim. Theory Appl. 69(1), 153–167 (1991)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Loridan, P.: \(\varepsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 43(2), 265–276 (1984)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Loridan, P.: \(\varepsilon \)-duality theorem of nondifferentiable nonconvex multiobjective programming. J. Optim. Theory Appl. 74(3), 565–566 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999)MATHGoogle Scholar
  20. 20.
    Ruhe, L.G., Fruhwirth, G.B.: \(\varepsilon \)-optimality for bicriteria programs and its application to minimum cost flows. Computing 44(1), 21–34 (1990)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Salz, W.: Eine topologische Eigenschaft der effizienten Punkte konvexer Mengen. Oper. Res. Verfahren XXIII. 23, 197–202 (1976)MathSciNetMATHGoogle Scholar
  22. 22.
    Sayadi-Bander, A., Pourkarimi, L., Basirzadeh, H.: On Convex Coradiant Set. (under review)Google Scholar
  23. 23.
    Steuer, R.E.: Multiple Criteria Optimization: Theory. Computation and Application. John Wiley and Sons, New York (1986)MATHGoogle Scholar
  24. 24.
    Tammer, C.: Ageneralization of Ekelands variational principle. Optimization 25, 129–141 (1992)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tanaka, T.: A new approach to approximation of solutions in vector optimization problems. In: Proceedings of APORS. World Scientific Publishing, Singapore, 497–504 (1995)Google Scholar
  26. 26.
    Valyi, I.: Approximate saddle-point theorems in vector optimization. J. Optim. Theory Appl. 55(3), 435–448 (1987)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    White, D.J.: Epsilon efficiency. J. Optim. Theory Appl. 49, 319–337 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Abbas Sayadi-bander
    • 1
  • Latif Pourkarimi
    • 2
  • Refail Kasimbeyli
    • 3
  • Hadi Basirzadeh
    • 1
  1. 1.Department of Mathematics, Faculty of Mathematical Sciences and ComputerShahid Chamran University of AhvazAhvazIran
  2. 2.Department of MathematicsRazi UniversityKermanshahIran
  3. 3.Department of Industrial EngineeringAnadolu UniversityEskisehirTurkey

Personalised recommendations