Strong Stationarity Conditions for a Class of Optimization Problems Governed by Variational Inequalities of the Second Kind
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We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primal–dual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional problems and for problems in function spaces, under suitable assumptions on the active set. A characterization of Bouligand and strong stationary points is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.
KeywordsVariational inequalities Optimality conditions Mathematical programs with equilibrium constraints
Mathematics Subject Classification49K21 90C33 35R35 49J40
The authors would like to thank Gerd Wachsmuth (TU Chemnitz) for his hint concerning strong stationarity. This work was supported by a DFG grant within the Collaborative Research Center SFB 708 (3D-Surface Engineering of Tools for Sheet Metal Forming Manufacturing, Modeling, Machining), which is gratefully acknowledged. We also acknowledge support of MATHAmSud project “Sparse Optimal Control of Differential Equations” (SOCDE).
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