Preconditioning for active set and projected gradient methods as semi-smooth Newton methods for PDE-constrained optimization with control constraints

Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the control poses a significant additional challenge for optimization methods. In this paper we propose preconditioners for the saddle point problems that arise when a primal-dual active set method is used. We also show for this method that the same saddle point system can be derived when the method is considered as a semi-smooth Newton method. In addition, the projected gradient method can be employed to solve optimization problems with simple bounds and we discuss the efficient solution of the linear systems in question. In the case when an acceleration technique is employed for the projected gradient method, this again yields a semi-smooth Newton method that is equivalent to the primal-dual active set method. Numerical results illustrate the competitiveness of this approach