Semismooth Newton Methods for Operator Equations in Function Spaces

We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCP-function-based reformulations of infinite-dimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and -order semismoothness from finite-dimensional spaces to a Banach space setting. Hereby, a new generalized differential is used that can be seen as an extension of Qi’s finite-dimensional C-subdifferential to our infinite-dimensional framework. We apply these semismoothness results to develop a Newton-like method for nonsmooth operator equations and prove its local q-superlinear convergence to regular solutions. If the underlying operator is -order semismoothness, convergence of q-order 1+ is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrating examples and by applications to nonlinear complementarity problems.