In this paper, we develop the notion of a phi-minimizer (or strict efficiency) for a set-valued map. Its properties and relations with other similar notions are studied. In special circumstances, under suitable conditions, we prove that a point is a phi-minimizer of a vector function if and only if it is a phi-minimizer of a certain family of linear scalarizations. We also establish a characterization of strict efficiency through a nonlinear scalarization, which is a generalization of the Gerstewitz function defined on the power set of the image space. The final part is focused on minimizers of order one, and we provide several necessary or sufficient conditions ( without convexity assumptions) through different kinds of derivatives as contingent, radial among others. Various illustrative examples showing the applicability of our results are also presented.