A Newton derivative scheme for shape optimization problems constrained by variational inequalities

Abstract

Shape optimization problems constrained by variational inequalities (VI) are non-smooth and non-convex optimization problems. The non-smoothness arises due to the variational inequality constraint, which makes it challenging to derive optimality conditions. Besides the non-smoothness there are complementary aspects due to the VIs as well as distributed, non-linear, non-convex and infinite-dimensional aspects due to the shapes which complicate to set up an optimality system and, thus, to develop fast and higher order solution algorithms. In this paper, we consider Newton-derivatives in order to formulate optimality conditions. In this context, we set up a Newton-shape derivative scheme. Examples show the application of the proposed scheme.