Bilevel Optimization of the Kantorovich Problem and its Quadratic Regularization Part III: The Finite-Dimensional Case

Abstract: 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.