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

arXiv: https://arxiv.org/abs/1711.02720 · HTML: https://mlsanity.com/paper/1711.02720/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2017.

## Built on by (most cited first, 1 of 1)

- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)

## Abstract (arXiv)

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). In contrast to classical results, our method of proof does not rely on Attouch's theorem on the characterization of Mosco convergence but 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.