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

Abstract

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 derive a necessary characterization of the elements of the Bouligand subdifferential of the solution mapping, together with a sufficient condition, in terms of admissible partitions of the biactive set, under which this characterization can be certified; as a consequence, the directional derivative is shown to admit a linear representative in every direction. We then derive optimality conditions characterizing Bouligand-stationary and strong-stationary points for discrete VI-constrained optimal control problems. Motivated by these results, a nonsmooth trust-region algorithm for solving the control problems is proposed, and a numerical experiment is presented as a proof of concept to illustrate the main properties of both the solution and the algorithm.