Necessary optimality conditions for optimization problems with variational inequality constraints

In this paper we study optimization problems with variational inequality constraints in finite dimensional spaces. Karush–Kuhn–Tucker type necessary optimality conditions involving coderivatives are given under certain constraint qualifications including one that ensures nonexistence of nontrivial abnormal multipliers. The result is applied to bilevel programming problems to obtain KKT type necessary optimality conditions. The KKT type necessary optimality conditions are shown to be satisfied without any constraint qualification by the class of bilevel programming problems where the lower level is a parametric linear quadratic problem.