, Volume 20, Issue 1, pp 99–114 | Cite as

A new approach to constrained optimization via image space analysis

  • M. ChinaieEmail author
  • J. Zafarani


In this article, by introducing a class of nonlinear separation functions, the image space analysis is employed to investigate a class of constrained optimization problems. Furthermore, the equivalence between the existence of nonlinear separation function and a saddle point condition for a generalized Lagrangian function associated with the given problem is proved.


Image space analysis Scalarization of vector optimization Linear and nonlinear separation Saddle point  Generalized Lagrangian 

Mathematics Subject Classification

90C26 90C29 26B25 



The authors are grateful to the referee for the comments and helpful suggestions, which have improved the presentation of this paper. The second author was partially supported by the Center of Excellence for Mathematics, University of Isfahan, Iran.


  1. 1.
    Benoist, J., Borwein, J.M., Popovici, N.A.: Characterization of quasiconvex vector- valued functions. Proc. Am. Math. Soc. 131, 1109–1113 (2001)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Benoist, J., Popovici, N.: Characterization of convex and quasiconvex set-valued maps. Math. Methods Oper. Res. 57, 427–435 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Castellani, G., Giannessi, F.: Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems. Survey of mathematical pro- gramming. In: Proceedings of ninth international mathematical programming symposium, Budapest, vol 2. North-Holland, pp. 423–439 (1979)Google Scholar
  4. 4.
    Chen, J., Li, S., Wan, Z., Yao, J. C.: Vector variational-like inequalities with constraints: separation and alternative. J. Optim. Theory Appl. (2015, in press)Google Scholar
  5. 5.
    Chinaie, M., Zafarani, J.: Image space analysis and scalarization of multivalued opti-mization. J. Optim. Theory Appl. 142, 451–467 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Chinaie, M., Zafarani, J.: Image space analysis and scalarization for \(\varepsilon \)-optimization of multifunctions. J. Optim. Theory Appl. 157, 685–695 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Dien, P.H., Mastroeni, G., Pappalardo, M., Quang, P.H.: Regularity condition for constrained extreme problems via image space. J. Optim. Theory Appl. 80, 19–37 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities. Image space analysis and separation.Vector variational inequalities and vector equilibria. In: Giannessi, F. (ed.) Mathematical Theories. Kluwer, Dordrecht (1999)Google Scholar
  10. 10.
    Giannessi, F., Pellegrini, L.: Image space analysis for vector optimization and variational inequalities. Scalarization. Combinatorial and Global Optimization, Ser. Appl. Math., vol. 14, pp. 97–110. World Sci. Publ., River Edge (2002)Google Scholar
  11. 11.
    Giannessi, F.: Constrained Optimization and Image Space Analysis. Separation of Sets and Optimality Conditions, vol. 1. Springer, New York (2005)Google Scholar
  12. 12.
    Giannessi, F., Maugeri, A.: Variational Analysis and Applications, Non Convex Opti- mization and Its Applications, vol. 79. Springer, New York (2005)Google Scholar
  13. 13.
    Giannessi, F., Mastroeni, G., Yao, J.-C.: On maximum and variational principles via image space analysis. Positivity 16, 405–427 (2012)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Jahn, J.: Vector Optimization. Theory, Applications and Extensions. Springer, Berlin (2011)zbMATHGoogle Scholar
  15. 15.
    Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical program- ming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Li, Z.F., Chen, G.Y.: Lagrangian multipliers, saddle points, and duality in vector optimization of set-valued maps. J. Math. Anal. Appl. 215, 297–316 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Li, J., Feng, S.Q., Zhang, Z.: A unified approach for constrained extremum problems: Image Space Analysis. J. Optim. Theory Appl. 159, 69–92 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)CrossRefGoogle Scholar
  19. 19.
    Pappalardo, M.: Image space approach to penalty methods. J. Optim. Theory Appl. 64, 141–152 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1806 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems I: Image Space Analysis. J. Optim. Theory Appl. 161, 738–762 (2014)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IsfahanIsfahanIran
  2. 2.Department of MathematicsSheikhbahaee University and University of IsfahanIsfahanIran

Personalised recommendations