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

Michael Ulbrich

Technische Universität München

Zentrum Mathematik

Lehrstuhl für Angewandte Mathematik und Mathematische Statistik

In: M. C. Ferris, O. L. Mangasarian, and J.-S. Pang, (eds.), Complementarity:

Applications, Algorithms and Extensions, Kluwer Academic Publishers, 2001.

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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 L^p 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.