  We consider optimization problems constrained by partial differential
equations (PDEs) with additional constraints placed on the solution of the
PDEs. We develop a general and versatile framework using infinite-valued
penalization functions and Clarke subgradients and apply this to problems with
box constraints as well as more general constraints arising in applications,
such as constraints on the average value of the state in subdomains. The
framework also allows for problems with discontinuous data in the constraints.
We present numerical results of this algorithm for the elliptic case and
compare with other state-constrained algorithms.
