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.

针对非光滑最优控制问题，我们提出了一种半光滑牛顿型方法。它的特殊功能是准牛顿法和半光滑牛顿法的结合。与半光滑的牛顿法相比，这降低了计算成本，同时保持了局部超线性收敛。该方法适用于希尔伯特空间问题，其目标是光滑函数，正则项和非光滑凸函数之和。在这项工作的理论部分，我们建立了该方法在无穷维设置中的局部超线性收敛，并讨论了该方法在受限于箱约束的情况下对热方程的稀疏最优控制的应用。我们验证了在该应用中满足局部超线性收敛的假设，并且证明了如果初始误差和问题数据允许，则收敛可以在比希尔伯特空间更强的范式中进行。在数值部分，我们将对两种最佳控制问题（包括磁共振成像中涉及Bloch方程的双线性控制的工程问题）的混合方法进行深入研究。我们使用这个问题来证明新方法能够解决非凸，非平滑的大规模现实问题。除其他外，该研究涉及网格独立性，全球化技术和有限内存方法。我们始终观察到，基于混合方法的算法在运行时比半光滑的牛顿算法快几倍。包括来自磁共振成像的工程问题，涉及到Bloch方程的双线性控制。我们使用这个问题来证明新方法能够解决非凸，非平滑的大规模现实问题。除其他外，该研究涉及网格独立性，全球化技术和有限内存方法。我们始终观察到，基于混合方法的算法在运行时比半光滑的牛顿算法快几倍。包括来自磁共振成像的工程问题，涉及到Bloch方程的双线性控制。我们使用这个问题来证明新方法能够解决非凸，非平滑的大规模现实问题。除其他外，该研究涉及网格独立性，全球化技术和有限内存方法。我们始终观察到，基于混合方法的算法在运行时比半光滑的牛顿算法快几倍。