Acta Mathematicae Applicatae Sinica, English Series

, Volume 19, Issue 3, pp 371–386

On First Order Optimality Conditions for Vector Optimization

  • L. M. Graña Drummond
  • A. N. Iusem
  • B. F. Svaiter
Original papers

DOI: 10.1007/s10255-003-0112-4

Cite this article as:
Graña Drummond, L.M., Iusem, A.N. & Svaiter, B.F. Acta Mathematicae Applicatae Sinica, English Series, English Series (2003) 19: 371. doi:10.1007/s10255-003-0112-4


We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented.


Cone constraints vector optimization Pareto minimization first order optimality conditions convex programming duality 

2000 MR Subject Classification

90C25 90C30 

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • L. M. Graña Drummond
    • 1
  • A. N. Iusem
    • 2
  • B. F. Svaiter
    • 2
  1. 1.Programa de Engenharia de Sistemas de Computação COPPE-UFRJ, CP 68511Rio de Janeiro-RJBrazil
  2. 2.Instituto de Matemática Pura e Aplicada (IMPA)Rio de Janeiro, RJBrazil