Abstract: The strong efficiency for set-valued optimization is considered in real normed spaces. With the help of the properties of Henig dilating cone and base functional, by applying generalized second-order cone-directed contingent derivates, a second-order optimality  necessary condition is established for a pair to be a strongly efficient element of set-valued optimization whose constraint condition is determined by  a set-valued mapping. When objective function  is nearly cone-subconvexlike, with the scalarization theorem for a strongly efficient point an optimality sufficient condition is also derived for a pair to be a strongly efficient element of set-valued optimization.

Key words: strong efficiency, generalized second-order cone-directed contingent deriv-ate, set-valued optimization