Abstract and Applied Analysis

Necessary and Sufficient Conditions for Set-Valued Maps with Set Optimization

Abdessamad Oussarhan and Ikram Daidai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Optimality conditions are studied for set-valued maps with set optimization. Necessary conditions are given in terms of S -derivative and contingent derivative. Sufficient conditions for the existence of solutions are shown for set-valued maps under generalized quasiconvexity assumptions.

Article information

Abstr. Appl. Anal., Volume 2018 (2018), Article ID 5962049, 6 pages.

Received: 15 August 2017
Accepted: 10 December 2017
First available in Project Euclid: 14 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Oussarhan, Abdessamad; Daidai, Ikram. Necessary and Sufficient Conditions for Set-Valued Maps with Set Optimization. Abstr. Appl. Anal. 2018 (2018), Article ID 5962049, 6 pages. doi:10.1155/2018/5962049. https://projecteuclid.org/euclid.aaa/1518577260

Export citation


  • T. Amahroq and A. Taa, “Sufficient conditions of optimality for multiobjective optimization problems with $\gamma $-paraconvex data,” Studia Mathematica, vol. 124, no. 3, pp. 139–247, 1997.
  • H. W. Corley, “Optimality conditions for maximizations of set-valued functions,” Journal of Optimization Theory and Applications, vol. 58, no. 1, pp. 1–10, 1988.
  • J. Jahn and A. A. Khan, “Generalized contingent epiderivatives in set-valued optimization: optimality conditions,” Numerical Functional Analysis and Optimization, vol. 23, no. 7-8, pp. 807–831, 2002.
  • A. A. Khan, C. Tammer, and C. Zalinescu, Set-valued optimization. Vector optimization, Springer, Heidelberg, Germany, 2015.
  • D. T. Luc, “Contingent derivatives of set-valued maps and applications to vector optimization,” Programs in Mathematics, vol. 50, pp. 99–111, 1991.
  • D. S. Shi, “Contingent derivative of the perturbation map in multiobjective optimization,” Journal of Optimization Theory and Applications, vol. 70, no. 2, pp. 385–396, 1991.
  • A. Taa, “Necessary and sufficient conditions for multiobjective optimization problems,” Optimization. A Journal of Mathematical Programming and Operations Research, vol. 36, no. 2, pp. 97–104, 1996.
  • D. Kuroiwa, “Some duality theorems of set-valued optimization with natural criteria,” in Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, pp. 221–228, 1999.
  • E. Hernández, L. Rodríguez-Marín, and M. Sama, “On solutions of set-valued optimization problems,” Computers & Mathematics with Applications, vol. 60, pp. 1401–1408, 2010.
  • M. Alonso-Durán and L. Rodríguez-Marín, “Optimality conditions for set-valued maps with set optimization,” Nonlinear Analysis, vol. 70, pp. 3057–3064, 2009. \endinput