We consider saddle point problems arising as (linearized) optimality conditions in elliptic optimal control problems. The efficient solution of such systems is a core ingredient in second-order optimization algorithms. In the spirit of Bramble and Pasciak, the preconditioned systems are symmetric and positive definite with respect to a suitable scalar product. We extend previous work by Schoeberl and Zulehner and consider problems with control and state constraints. It stands out as a particular feature of this approach that an appropriate symmetric indefinite preconditioner can be constructed from standard preconditioners for those matrices which represent the inner products, such as multigrid cycles.
Numerical examples in 2D and 3D are given which illustrate the performance of the method, and limitations and open questions are addressed.