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Shape optimization problems constrained by variational inequalities (VI) arenon-smooth and non-convex optimization problems. The non-smoothness arises dueto the variational inequality constraint, which makes it challenging to deriveoptimality conditions. Besides the non-smoothness there are complementaryaspects due to the VIs as well as distributed, non-linear, non-convex andinfinite-dimensional aspects due to the shapes which complicate to set up anoptimality system and, thus, to develop efficient solution algorithms. In thispaper, we consider the Hadamard semiderivative in order to formulate optimalityconditions. In this context, we set up a Hadamard shape semiderivative approachand demonstrate its advantages over an proposed approach based onregulatizations.

A semiderivative approach for shape optimization problems constrained by variational inequalities