Bilevel Optimal Transport Problems: Existence, Regularization and Convergence

In this thesis we consider bilevel optimization problems of the form min{R(x) + P(y) | x solves a transport problem that depends on y}, where y either prescribes the source distribution that is to be transported or parametrizes the cost of transportation. We consider two different formulations for the transport problem in the lower level. The Kantorovich Problem asks to find a measure on the cartesian product of two sets such that it minimizes a certain linear cost functional while the marginals of that measure must match a given source and target distribution. The Beckmann Problem asks to find a vector field such that it again minimizes a certain cost functional and its divergence matches the difference of a prescribed source distribution and a prescribed target distribution on a given set. Both problem formulations come from the field of Optimal Transport. Solutions are usually not unique for either of them and in general do not admit densities w. r. t. the Lebesgue measure, which makes it challenging to treat them in the bilevel setting above. To ensure uniqueness of solutions and improve regularity properties, we regularize both the Kantorovich and the Beckmann problem with an additive term. We prove existence of solutions for both regularized lower level problems under suitable assumptions. Existence of solutions for the bilevel problem is established only for the Beckmann problem in the lower level, but for both the (doubly) regularized and the unregularized variant. Apart from existence of solutions, our main interest is to analyze how the regularized (bilevel) problems behave for vanishing regularization. More precisely, we aim to show that solutions of the regularized problems converge towards solutions of the unregularized problems in a suitable sense. We address this question for both the Beckmann and the Kantorovich problem, while in the bilevel setting we focus on the case where the Beckmann problem is used as lower level problem.