Advertisement

Journal of Optimization Theory and Applications

, Volume 180, Issue 2, pp 428–441 | Cite as

Subdifferential Calculus for Set-Valued Mappings and Optimality Conditions for Multiobjective Optimization Problems

  • Ahmed TaaEmail author
Article
  • 83 Downloads

Abstract

In this work, we provide a generalized formula for the weak subdifferential (resp., for the Benson proper subdifferential) of the sum of two cone-closed and cone-convex set-valued mappings, under the Attouch–Brézis qualification condition. This formula is applied to establish necessary and sufficient optimality conditions in terms of Lagrange/Karush/Kuhn/Tucker multipliers for the existence of the weak (resp., of the Benson proper) efficient solutions of a set-valued vector optimization problem.

Keywords

Set-valued vector optimization Subdifferential Optimality conditions Lagrange/Karush/Kuhn/Tucker multipliers 

Mathematics Subject Classification

90C29 90C26 90C46 

Notes

Acknowledgements

The author thanks the anonymous referee and the Editor Hedy Attouch for their helpful remarks that allowed us to improve the original presentation.

References

  1. 1.
    Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, New York (1985)zbMATHGoogle Scholar
  2. 2.
    Tanino, T.: Conjugate duality in vector optimization. J. Math. Anal. Appl. 167, 84–97 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Sawaragi, Y., Tanino, T.: Conjugate maps and duality in multiobjective optimization. J. Optim. Theory Appl. 31, 473–499 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Taa, A.: Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization problems. J. Math. Anal. Appl. 283, 398–415 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sach, P.H.: Moreau–Rockafellar theorems for nonconvex set-valued maps. J. Optim. Theory Appl. 133, 213–227 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Taa, A.: On subdifferential calculus for set-valued mappings and optimality conditions. Nonlinear Anal. 74, 7312–7324 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Taa, A.: \(\varepsilon \)-Subdifferentials of set-valued maps and \(\varepsilon \)-weak Pareto optimality for multiobjective optimization. Math. Methods Oper. Res. 62, 187–209 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Tuan, L.A.: \(\varepsilon \)-Optimality conditions for vector optimization problems with set-valued maps. Numer. Funct. Anal. Optim. 31, 78–95 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  10. 10.
    Attouch, H., Brézis, H.: Duality for the sum of convex functions in general Banach spaces. In: Baroso, J.A. (ed.) Aspects of Mathematics and it Applications, pp. 125–133. Elsevier, Amsterdam (1986)CrossRefGoogle Scholar
  11. 11.
    Rodrigues, B., Simons, S.: Conjugate functions and subdifferentials in nonnormed situations for operators with complete graphs. Nonlinear Anal. 12, 1069–1078 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Göpfert, A., Tammer, C., Riahi, H., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)zbMATHGoogle Scholar
  13. 13.
    Dauer, J.P., Saleh, O.A.: A characterization of proper minimal points as solutions of sublinear optimization problems. J. Math. Anal. Appl. 178, 227–246 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de MathématiquesFaculté des Sciences et Techniques de MarrakechMarrakechMorocco

Personalised recommendations