Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control
Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control
复制标题
障碍型变分不等式解算子的半光滑性及其在最优控制中的应用
DOI：
--
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
G. Wachsmuth
中科院分区：
文献类型：
--
作者：
C. Christof;G. Wachsmuth
证明了椭圆型障碍型变分不等式（或更一般地，具有某些逐点a.e.凸性质）是牛顿可微的，当被认为是适当的Lebesgue空间之间的映射，并配备了强-弱Bouligand微分作为广义集值导数。它表明，这种牛顿可微性允许解决最优控制问题的H1-成本条款和单边逐点控制约束的半光滑牛顿方法。在无限维情况下证明了算法的超线性收敛性，并在数值实验中证明了算法的网格独立性。我们期望本文的研究结果也有助于拟变分不等式数值解程序的设计和障碍型变分问题的最优控制。
We prove that solution operators of elliptic obstacle-type variational inequalities (or, more generally, locally Lipschitz continuous functions possessing certain pointwise-a.e. convexity properties) are Newton differentiable when considered as maps between suitable Lebesgue spaces and equipped with the strong-weak Bouligand differential as a generalized set-valued derivative. It is shown that this Newton differentiability allows to solve optimal control problems with H1-cost terms and one-sided pointwise control constraints by means of a semismooth Newton method. The superlinear convergence of the resulting algorithm is proved in the infinite-dimensional setting and its mesh independence is demonstrated in numerical experiments. We expect that the findings of this paper are also helpful for the design of numerical solution procedures for quasi-variational inequalities and the optimal control of obstacle-type variational problems.