Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points

Abstract

In this paper, we consider a class of constrained vector optimization problems by using image space analysis. A class of vector-valued separation functions and a \begin{document}$ \mathfrak{C} $\end{document} -solution notion are proposed for the constrained vector optimization problems, respectively. Moreover, existence of a saddle point for the vector-valued separation function is characterized by the (regular) separation of two suitable subsets of the image space. By employing the separation function, we introduce a class of generalized vector-valued Lagrangian functions without involving any elements of the feasible set of constrained vector optimization problems. The relationships between the type-Ⅰ(Ⅱ) saddle points of the generalized Lagrangian functions and that of the function corresponding to the separation function are also established. Finally, optimality conditions for \begin{document}$ \mathfrak{C} $\end{document} -solutions of constrained vector optimization problems are derived by the saddle-point conditions.