Variational inequalities and optimization problems

Abstract

In this paper we survey the relationships between scalar and vector variational inequalities (of differential type) and the underlying optimizationproblem. We show that the variational inequalities of Stampacchia type can be viewed as necessary optimality conditions, while the variational inequalities of Minty type can be considered as sufficient optimality conditions. Concerning this last statement, a gap is observed between the scalar and the vector case and possible fulfilments of this gap are investigated