Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators

  • Constantin ChristofEmail author
  • Gerd Wachsmuth


This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem.


Variational inequalities Sensitivity analysis Directional differentiability Bang–bang Optimal control Differential stability Second-order epi-differentiability 

Mathematics Subject Classification

90C31 49K40 47J20 



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Authors and Affiliations

  1. 1.Faculty of MathematicsTechnische Universität MünchenGarchingGermany
  2. 2.Chair of Optimal Control, Institute of MathematicsBrandenburgische Technische Universität Cottbus-SenftenbergCottbusGermany

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