  Shape optimization problems constrained by variational inequalities (VI) are
non-smooth and non-convex optimization problems. The non-smoothness arises due
to the variational inequality constraint, which makes it challenging to derive
optimality conditions. Besides the non-smoothness there are complementary
aspects due to the VIs as well as distributed, non-linear, non-convex and
infinite-dimensional aspects due to the shapes which complicate to set up an
optimality system and, thus, to develop efficient solution algorithms. In this
paper, we consider G\^ateaux semiderivatives in order to formulate optimality
conditions. In the application, we concentrate on a shape optimization problem
constrained by the contact problem.
