Nonsmooth and Nonconvex Optimal Transport Problems
In recent years a strong interest has developed within mathematics in so-called "branched Transport" models, which allow to describe transportation networks as they occur in road systems, river basins, communication networks, vasculature, and many other natural and artificial contexts. As in classical optimal transport, an amount of material needs to be transported efficiently from a given initial to a final mass distribution. In branched transport, however, the transportation cost is not proportional, but subadditive in the transported mass, modelling an increased transport efficiency if mass is transported in bulk. This automatically favours transportation schemes in which the mass flux concentrates on a complicated, ramified network of one-dimensional lines. The branched transport problem is an intricate nonconvex, nonsmooth variational problem on Radon measures (in fact on normal currents) that describe the mass flux. Various different formulations were developed and analysed (including work by the PIs), however, they all all take the viewpoint of geometric measure theory, working with flat chains, probability measures on the space of Lipschitz curves, or the like. What is completely lacking is an optimization and optimal control perspective (even though some ideas of optimization shimmer through in the existing variational arguments such as regularity analysis via necessary optimality conditions or the concept of calibrations which are related to dual optimization variables). This situation is also reflected in the fact that the field of numerics for branched transport is rather underdeveloped and consists of ad hoc graph optimization methods for special cases and two-dimensional phase field approximations. We will reformulate branched transport in the framework of optimization and optimal control for Radon measures, work out this optimization viewpoint in the variational analysis of branched transport networks, and exploit the results in novel numerical approaches. The new perspective will at the same time help variational analysts, advance the understanding of nonsmooth, nonconvex optimization problems on measures, and provide numerical methods to obtain efficient transport networks.
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Christian Clason, Dirk A. Lorenz, Hinrich Mahler, Benedikt Wirth:
Entropic Regularization of Continuous Optimal Transport Problems