  In general, standard necessary optimality conditions cannot be formulated in
a straightforward manner for semi-smooth shape optimization problems. In this
paper, we consider shape optimization problems constrained by variational
inequalities of the first kind, so-called obstacle-type problems. Under
appropriate assumptions, we prove existence of adjoints for regularized
problems and convergence to limiting objects of the unregularized problem.
Moreover, we derive existence and closed form of shape derivatives for the
regularized problem and prove convergence to a limit object. Based on this
analysis, an efficient optimization algorithm is devised and tested
numerically.
