Applications of optimal transport problems occur in several areas such as imaging, economy or machine learning. The goal of optimal transport is, to map a given mass distribution to a target distribution in a way that a given objective function for the transport cost is minimal. In this project we will consider bilevel optimization problems where the optimal transport problems occur as constraints. The upper level optimization variables can be the source mass distribution or the weights in the lower level transport problem. Possible applications of such bilevel problems include mass identification problems and congestion minimization problems.

This project focuses on two formulations of optimal transport, namely the Beckmann and Kantorovich formulation (which are dual to each other under certain assumptions). For both problems solution are not necessarily unique and additionally solutions in general have low regularity. To counteract these challenges in the context of bilevel optimization we will use tailored regularization methods which keep essential non-smooth features such as sparsity of the solutions. Besides a convergence analysis of the regularization methods we will develop efficient non-smooth optimization algorithms for the regularized bilevel problems. These algorithms shall be use to solve the aforementioned problems of mass identification or congestion minimization.