November 8, 2025Open Access

Nonlinear Dynamical Unbalanced Optimal Transport: Relaxation and Duality

Key Points

Non-convex optimization transforms into a convex dual formulation using techniques from dual optimal control.
A first-order algorithm based on the dual formulation aims to solve the problem numerically, enhancing tractability.
The study incorporates mass dissipation and limited control input in the framework of optimal transport.
This approach highlights the significance of HJB equations in deriving smooth sub-solutions to the transport problem.

Abstract

In this paper, we introduce a generalized dynamical unbalanced optimal transport framework by incorporating limited control input and mass dissipation, addressing limitations in conventional optimal transport for control applications. We derive a convex dual of the problem using dual optimal control techniques developed before and during the 1990s, transforming the non-convex optimization into a more tractable form.At the core of this formulation is the smooth sub-solutions to an HJB equation. A first-order algorithm based on the dual formulation is proposed to solve the problem numerically.

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