Set-Optimization Meets Variational Inequalities

We study necessary and sufficient conditions to attain solutions of set-optimization problems in terms of variational inequalities of Stampacchia and Minty type. The notion of solution we deal with has been introduced in [18]. 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 scalar problems. The optimality conditions are given by Stampacchia and Minty type Variational inequalities, defined both by set-valued directional derivatives and by Dini derivatives of the scalarizations. The main results allow to obtain known variational characterizations for vector-valued optimization problems as special case.