Bilevel Optimal Transport
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.
Christian Clason, Dirk A. Lorenz, Hinrich Mahler, Benedikt Wirth: Entropic Regularization of Continuous Optimal Transport Problems, Journal of Mathematical Analysis and Applications, 2021 (SPP1962-121).
Dirk A. Lorenz, Paul Manns, Christian Meyer: Quadratically Regularized Optimal Transport, Applied Mathematics & Optimization, 2019 (SPP1962-120).
Dirk Lorenz, Hinrich Mahler: Orlicz Space Regularization of Continuous Optimal Transport Problems (SPP1962-161, 03/2021, [bib])
Christian Clason, Dirk A. Lorenz, Hinrich Mahler, Benedikt Wirth: Entropic Regularization of Continuous Optimal Transport Problems (SPP1962-121, 09/2019, [bib])
Modeling, problem analysis, algorithm design and convergence analysisThe focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.
Project Related News
Jan 19, 2022 : New preprint submitted
Hinrich Mahler submitted the preprint SPP1962-187 $L^alpha$-Regularization of the Beckmann Problem.
Mar 25, 2021 : New preprint submitted
Hinrich Mahler submitted the preprint SPP1962-161 Orlicz Space Regularization of Continuous Optimal Transport Problems.
Sep 23, 2019 : New preprint submitted
Dirk A. Lorenz submitted the preprint SPP1962-121 Entropic Regularization of Continuous Optimal Transport Problems.
Sep 16, 2019 : New preprint submitted
Christian Meyer submitted the preprint SPP1962-120 Quadratically Regularized Optimal Transport.