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

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Abstract

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.

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Authors and Affiliations

  1. Faculty of Mathematics, Technische Universität München, 85748, Garching, Germany

    Constantin Christof

  2. Chair of Optimal Control, Institute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, 03046, Cottbus, Germany

    Gerd Wachsmuth

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  1. Constantin Christof
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  2. Gerd Wachsmuth
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Correspondence to Constantin Christof.

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This research was supported by the German Research Foundation (DFG) under Grant Numbers ME 3281/7-1 and WA 3636/4-1 within the priority program “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” (SPP 1962).

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Christof, C., Wachsmuth, G. Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators. Appl Math Optim 81, 23–62 (2020). https://doi.org/10.1007/s00245-018-09553-y

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