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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 G\^ateaux semiderivatives in order to formulate optimalityconditions. In the application, we concentrate on a shape optimization problemconstrained by the contact problem.

G\^ateaux semiderivative approach applied to shape optimization for contact problems