On variational inequalities using directional convexificators

In this paper, we give some results which constitute an application of directional convexificators recently introduced by Dempe and Pilecka [Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Global Optim. 2015;61:769–788]. After establishing mean value conditions in terms of directional convexificators, we formulate variational inequalities of Stampacchia and Minty type in terms of directional convexificators and use these variational inequalities as a tool to find out necessary and sufficient conditions for a point to be an optimal solution of an inherent optimization problem. An example illustrating our findings is also given.