On a Nonsmooth Newton Method for Nonlinear Complementarity Problems in Function Space with Applications to Optimal Control

Many applications in mathematical modeling and optimal control lead to problems that are posed in function spaces and contain pointwise complementarity conditions. In this paper, a projected Newton method for nonlinear complementarity problems in the infinite dimensional function space Lp is proposed and analyzed. Hereby, an NCP-function is used to reformulate the problem as a nonsmooth operator equation. The method stays feasible with respect to prescribed bound-constraints. The convergence analysis is based on semismoothness results for superposition operators in function spaces. The proposed algorithm is shown to converge locally q-superlinearly to a regular solution. As an important tool for applications, we establish a sufficient condition for regularity. The application of the algorithm to the distributed bound-constrained control of an elliptic partial differential equation is discussed in detail. Numerical results confirm the efficiency of the method.