Dynamic Models of Wasserstein-1-Type Unbalanced Transport
This work provides theoretical foundations and computational simplifications for unbalanced optimal transport, benefiting researchers in optimal transport and its applications.
The paper derives an equivalent static formulation for a class of dynamic unbalanced transport models with transport costs proportional to distance, enabling more efficient computation, and analyzes the properties of optimal solutions and their relation to existing models.
We consider a class of convex optimization problems modelling temporal mass transport and mass change between two given mass distributions (the so-called dynamic formulation of unbalanced transport), where we focus on those models for which transport costs are proportional to transport distance. For those models we derive an equivalent, computationally more efficient static formulation, we perform a detailed analysis of the model optimizers and the associated optimal mass change and transport, and we examine which static models are generated by a corresponding equivalent dynamic one. Alongside we discuss thoroughly how the employed model formulations relate to other formulations found in the literature.