A hybrid semismooth quasi-Newton method for nonsmooth optimal control with PDEs

We propose a semismooth Newton-type method for nonsmooth optimal control problems. Its particular feature is the combination of a quasi-Newton method with a semismooth Newton method. This reduces the computational costs in comparison to semismooth Newton methods while maintaining local superlinear convergence. The method applies to Hilbert space problems whose objective is the sum of a smooth function, a regularization term, and a nonsmooth convex function. In the theoretical part of this work we establish the local superlinear convergence of the method in an infinite-dimensional setting and discuss its application to sparse optimal control of the heat equation subject to box constraints. We verify that the assumptions for local superlinear convergence are satisfied in this application and we prove that convergence can take place in stronger norms than that of the Hilbert space if initial error and problem data permit. In the numerical part we provide a thorough study of the hybrid approach on two optimal control problems, including an engineering problem from magnetic resonance imaging that involves bilinear control of the Bloch equations. We use this problem to demonstrate that the new method is capable of solving nonconvex, nonsmooth large-scale real-world problems. Among others, the study addresses mesh independence, globalization techniques, and limited-memory methods. We observe throughout that algorithms based on the hybrid methodology are several times faster in runtime than their semismooth Newton counterparts.