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We prove that solution operators of elliptic obstacle-type variationalinequalities (or, more generally, locally Lipschitz continuous functionspossessing certain pointwise-a.e. convexity properties) are Newtondifferentiable when considered as maps between suitable Lebesgue spaces andequipped with the strong-weak Bouligand differential as a generalizedset-valued derivative. It is shown that this Newton differentiability allows tosolve optimal control problems with H1-cost terms and one-sided pointwisecontrol constraints by means of a semismooth Newton method. The superlinearconvergence of the resulting algorithm is proved in the infinite-dimensionalsetting and its mesh independence is demonstrated in numerical experiments. Weexpect that the findings of this paper are also helpful for the design ofnumerical solution procedures for quasi-variational inequalities and theoptimal control of obstacle-type variational problems.

Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control