  We consider an optimal control problem in which the state is governed by an
unilateral obstacle problem (with obstacle from below) and restricted by a
pointwise state constraint (from above). In the presence of control
constraints, we prove, via regularization of the state constraints, that a
system of C-stationarity is necessary for optimality. In the absence of control
constraints, we show that local minimizers are even strongly stationary by a
careful discussion of the primal first-order conditions of B-stationary type.
