Bilevel Optimal Transport Problems: Existence, Regularization and Convergence

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This thesis explores bilevel optimization problems structured as min{R(x) + P(y) | x solves a transport problem dependent on y}, where y influences the source distribution or the transportation cost. Two formulations of the transport problem are examined: the Kantorovich Problem, which seeks a measure on the Cartesian product of two sets to minimize a linear cost functional while ensuring the marginals match a specified source and target distribution, and the Beckmann Problem, which aims to find a vector field minimizing a cost functional with its divergence matching the difference between a given source and target distribution. Both formulations stem from Optimal Transport theory, but solutions are typically non-unique and may not possess densities with respect to the Lebesgue measure, complicating their treatment in the bilevel context. To enhance solution uniqueness and regularity, both problems are regularized with an additive term. The thesis establishes the existence of solutions for the regularized lower-level problems under appropriate conditions. However, existence for the bilevel problem is confirmed only for the Beckmann formulation, applicable to both regularized and unregularized cases. Additionally, the study investigates the behavior of regularized problems as regularization diminishes, demonstrating convergence of solutions from regularized to unregularized problems for both formulations, with a focus on the Be