  We study necessary and sufficient conditions to attain solutions of
set-optimization problems in therms of variational inequalities of Stampacchia
and Minty type. The notion of a solution we deal with has been introduced Heyde
and Loehne, for convex set-valued objective functions. To define the set-valued
variational inequality, we introduce a set-valued directional derivative and we
relate it to the Dini derivatives of a family of linearly scalarized problems.
The optimality conditions are given by Stampacchia and Minty type Variational
inequalities, defined both by the set valued directional derivative and by the
Dini derivatives of the scalarizations. The main results allow to obtain known
variational characterizations for vector valued optimization problems.
