Necessary and sufficient conditions for weak efficiency in non-smooth vectorial optimization problems

Abstract

By using the concepts of local cone approximations and K-directional derivatives, we obtain necessary conditions of optimality for the weak efficiency of the vectorial optimization problems with the inequalities and abstract constraints. We introduce the notion of stationary point (weak and strong) and we prove that, under suitable hypotheses of K-invexity, the stationary points are weakly efficient solutions (global).

Keywords:

• non-smooth vectorial optimization
• cone approximate
• directional generalized derivative
• weak efficient solution
• K-invex function

AMS 2000 Mathematics Subject Classifications: :

• 90C26
• 90C29
• 90C30
• 90C46
• 90C48

Acknowledgements

The third and fourth authors were supported by Ministerio De Ciencia y Tecnología, Spain, Grant BFM 2003-06579. We thank the anonymous referee for his valuable comments on the final version of the article and for indicating these references 17,21–23,27,31,34.