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State Constrained Optimization with Partial Differential Equations via Generalized Gradients

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.