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

TL;DR

This paper studies a quadratic regularization approach to bilevel optimal transport problems governed by the Kantorovich formulation, focusing on finite-dimensional cases to improve numerical properties and solution sparsity.

Contribution

It extends previous infinite-dimensional results to finite-dimensional cases and advances the numerical treatment of bilevel optimal transport problems with quadratic regularization.

Findings

Reproduction of well-posedness results in finite-dimensional setting

Demonstration of improved numerical properties with quadratic regularization

Foundation laid for numerical algorithms for bilevel problems

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.