Set optimization meets variational inequalities
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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 by Heyde and Löhne in 2011. 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 cases.