Bouligand Analysis and Discrete Optimal Control of Total Variation-Based Variational Inequalities

We investigate differentiability and subdifferentiability properties of the solution mapping associated with variational inequalities (VI) of the second kind involving the discrete total-variation. Bouligand differentiability of the solution operator is established via a direct quotient analysis applied to a primal-dual reformulation of the VI. By exploiting the structure of the directional derivative and introducing a suitable subspace, we fully characterize the Bouligand subdifferential of the solution mapping. We then derive optimality conditions characterizing Bouligand-stationary and strongly-stationary points for discrete VI-constrained optimal control problems. A trust-region algorithm for solving these control problems is proposed based on the obtained characterizations, and a numerical experiment is presented to illustrate the main properties of both the solution and the proposed algorithm.