By exploring the ideas around the Dubovitskii-Milyutin approach, necessary optimality conditions are given for various optimality notions in set-valued optimization. These optimality conditions are given by using the contingent derivative and the generalized contingent epiderivative of the objective set-valued map and the set-valued maps de. ning the constraints. The notions of subgradients and scalarized subgradients for set-valued maps are proposed and used to state some regularity conditions.