Set optimization meets variational inequalities

Abstract

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 that we relate to Dini derivatives of a family of scalar problems. 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 optimization problems as special cases.