A Hybrid Semismooth Quasi-Newton Method for Structured Nonsmooth Operator Equations in Banach Spaces

We present an algorithm for the solution of structured nonsmooth operator equations in Banach spaces. Specifically, we seek roots of mappings that involve the composition of a smooth outer and a semismooth inner map. To exploit this structure we propose a hybrid approach in which the semismooth part is linearized in the same way as in semismooth Newton methods while the smooth part is handled by a Broyden-like method. The resulting algorithm is a semismooth Newton-type method that does not require the evaluation of the derivative of the smooth part. We prove local q-linear and q-superlinear convergence results for the hybrid algorithm. In particular, this is the first work that establishes superlinear convergence of a semismooth quasi-Newton method in an infinite-dimensional setting. The convergence results also extend known finite-dimensional ones in that the structure of the equation and the algorithm under consideration are more general than those available in the literature. In addition, it is shown that q-linear convergence of the iterates and compactness of the initial operator discrepancy of the smooth part implies q-superlinear convergence without the assumption that the initial operator discrepancy is small in norm, which is a new type of result for semismooth quasi-Newton methods. The convergence theory is developed under mild assumptions, which yields extensions of available results for semismooth quasi-Newton methods as well as for Broyden-like methods. The benefit of the method in practical applications is addressed in a complementary paper. There, we show on problems from optimal control that the assumptions for q-superlinear convergence are satisfied and that the hybrid approach leads to highly competitive numerical schemes that have substantially lower runtimes than state-of-the-art semismooth Newton methods.