Motivated by some ideas in the work by Dinh The Luc [“Theory of vector optimization” (Lecture Notes Econ.Math.Sci.319; Berlin: Springer-Verlag) (1988; Zbl 0654.90082)] and by some subsequent papers by Benoist, Borwein, and Popovici, the authors characterize in terms of scalar preinvexity (resp.,prequasiinvexity) the set-valued functions which are \(K\)-preinvex (resp.,\(K\)-prequasiinvex) with respect to a closed convex cone \(K\). More precisely, in Section 3 they provide a characterization of a \(K\)-pre(quasi)invex set-valued function \(F:U\to 2^Y\), where \(U\) is an invex subset of a vector space, in terms of pre(quasi)-invexity of certain extended real-valued functions arising as the composition \(\ell\circ F\), where \(\ell\) is an extreme direction of the nonnegative polar cone of \(K\). Moreover, under stronger assumptions, they prove a necessary and sufficient condition for an upper hemicontinuous, and for a lower hemicontinuous set-valued map to be \(K\)-preinvex. In Section 4, they develop the same arguments for \(K\)-prequasiinvex set valued maps. The aim of Section 5 is to give sufficient conditions for a weakly \(K\)-preinvex (resp.,weakly \(K\) prequasiinvex) set-valued map to be \(K\)-preinvex (resp.,\(K\) prequasiinvex); these conditions go through the notion of \(\eta\)-segmentary epi-closedness. The last section is devoted to some results comparing local solutions and global solutions of a set-valued scalar optimization problem, and of a vector optimization problem with set-valued maps.