Optimality conditions of first and second order in vector optimization problems on metric spaces

For mappings defined on metric spaces with values in Banach spaces, the notions of derivative vectors of first and second order are introduced. These notions are used to establish necessary conditions and sufficient conditions of first and second order for points of local $\prec$-minimum of such mappings, where $\prec$ is the strict preorder relation defined on the space of values of the mapping that is minimized. Minimality conditions are obtained as corollaries for the case when the mapping is defined on a subset of a normed space.