We consider optimal control problems governed by an elliptic variational inequality of the first kind, namely the obstacle problem. The variational inequality is treated by penalization which leads to optimization problems governed by a nonsmooth semi- linear elliptic PDE. The CALi algorithm is then applied for the efficient solution of these nonsmooth optimization problems. The special feature of the optimization algorithm CALi is the treatment of the nonsmooth Lipschitz-continuous operators abs, max and min, which allows to explicitly exploit the nonsmooth structure. Stationary points are located by appropriate decomposition of the optimization problem into so called smooth compact abs-linearized problems. Each of these compact abs-linearized problems can be solved by classical means. The comprehensive algorithmic concept is presented and its performance is discussed through examples.
Institut für Mathematik, Humboldt-Universität zu Berlin 2: Institut für Mathematik und Computergestützte Simulation, Universität der Bundeswehr München