Entropic Proximal Gradient Method for Generalized Optimal Transport Problems

Abstract

Optimal transport, a fundamental problem in applied mathematics, involves finding the most efficient way to move mass from multiple sources to multiple destinations. Previously known approaches employ entropic regularization combined with the Sinkhorn iterations, a technique known for its efficiency in solving large-scale optimal transport problems. This thesis presents a new method for solving generalized optimal transport problems using the entropic proximal gradient method. The method breaks down the complex problem into a sequence of standard optimal transport problems, solved by the Sinkhorn iterations. We provide theoretical foundations, including proof of convergence and termination criteria, along with a detailed implementation and numerical experiments showing the algorithm’s applicability. The results of this thesis may offer improvements in computational performance for generalized optimal transport problems, making it a valuable tool for applications in economics, machine learning, and other fields where optimal transport is utilized.