  As the title suggests, this is the third paper in a series addressing bilevel
optimization problems that are governed by the Kantorovich problem of optimal
transport. These tasks can be reformulated as mathematical problems with
complementarity constraints in the space of regular Borel measures. Due to the
nonsmoothness that is introduced by the complementarity constraints, such
problems are often regularized, for instance, using entropic regularization. In
this series of papers, however, we apply a quadratic regularization to the
Kantorovich problem. By doing so, we enhance its numerical properties while
preserving the sparsity structure of the optimal transportation plan as much as
possible. While the first two papers in this series focus on the well-posedness
of the regularized bilevel problems and the approximation of solutions to the
bilevel optimization problem in the infinite-dimensional case, in this paper,
we reproduce these results for the finite-dimensional case and present findings
that go well beyond the ones of the previous papers and pave the way for the
numerical treatment of the bilevel problems.
