An Unbalanced Optimal Transport Problem with a Growth Constraint

Abstract

In this paper, we introduce several unbalanced optimal transport problems between two Radon measures with different total masses. Initially, we explore a generalization of the Benamou-Brenier problem, incorporating a growth constraint to accommodate a non-decreasing total mass during transportation. This leads to the formulation of a modified Hellinger-Kantorovich (mHK) problem. Our investigation reveals quasi-metric properties of this novel problem and characterizes it within a cone setting through a newly defined quasi-cone metric, resulting in an equivalent formulation of the mHK problem. This formulation simplifies the demonstration of the existence of optimal solutions and facilitates explicit calculations for transport problems between two Dirac measures.A significant advancement in our work is the construction of a dual problem for the mHK problem, a topic previously unexplored. We confirm the duality and identify optimality conditions for transport plans, successfully deriving a one-to-one (Monge) map under certain regularity conditions for the initial measure. Furthermore, we propose a dynamic formulation for the mHK problem within a cone setting, focusing on minimization over dynamic plans involving absolutely continuous curves between cone points. This approach not only projects a dynamic plan onto an absolutely continuous curve between initial and target measures but also establishes a close relationship with solutions to continuity equations.Motivated by dynamic models of biological growth, our study extends to practical applications, providing an equivalent convex formulation of the mHK problem and developing numerical schemes based on the Douglas-Rachford algorithm and the Alternating Direction Method of Multipliers algorithm. We apply these schemes to synthetic data, demonstrating the utility of our theoretical findings.