Nonsmooth optimization by successive abs-linearization in function spaces

We present and analyze the solution of nonsmooth optimization problems by a quadratic overestimation method in a function space setting. Under certain assumptions on a suitable local model, we show convergence to first-order minimal points. Subsequently, we discuss an approach to generate such a local model using the so-called abs-linearization. Finally, we discuss a class of PDE-constrained optimization problems incorporating the L-1-penalty term that fits into the considered class of nonsmooth optimization problems.