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We investigate optimality conditions for optimization problems constrained bya class of variational inequalities of the second kind. Based on a nonsmoothprimal-dual reformulation of the governing inequality, the differentiability ofthe solution map is studied. Directional differentiability is proved both forfinite-dimensional and function space problems, under suitable assumptions onthe active set. A characterization of B- and strong stationary optimalsolutions is obtained thereafter. Finally, based on the obtained first-orderinformation, a trust-region algorithm is proposed for the solution of theoptimization problems.

Strong stationarity conditions for a class of optimization problems governed by variational inequalities of the 2nd kind