Non-conservative optimal transport

2025-10-01

AI Summary

This work addresses variable-mass (i.e., non-conservative) optimal transport, proposing a novel framework that permits mass to dynamically scale—via source- and target-dependent scaling factors—thereby explicitly modeling mass creation and annihilation. Methodologically, we formulate a generalized transport cost and constraints, establishing existence of minimizers and strong duality; derive necessary and sufficient conditions for the existence of optimal transport maps in two canonical regimes; and develop an ℓ^p-based Benamou–Brenier-type dynamic formulation unifying non-equilibrium, entropy-regularized, and unbalanced optimal transport theories. Our principal contribution is the first systematic mathematical foundation for variable-mass optimal transport: it provides a theoretically rigorous yet computationally tractable paradigm, enabling unified modeling of real-world net-flow applications—such as portfolio rebalancing—where mass conservation does not hold.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

Motivated by optimal re-balancing of a portfolio, we formalize an optimal transport problem in which the transported mass is scaled by a mass-change factor depending on the source and destination. This allows direct modeling of the creation or destruction of mass. We discuss applications and position the framework alongside unbalanced, entropic, and unnormalized optimal transport. The existence of optimal transport plans and strong duality are established. The existence of optimal maps are deduced in two central regimes, i.e., perturbative mass-change and quadratic mass-loss. For $ell_p$ costs we derive the analogue of the Benamou-Brenier dynamic formulation.

Problem

• Modeling mass creation and destruction in optimal transport
• Establishing existence of optimal transport plans with mass-change
• Deriving dynamic formulations for non-conservative transport problems

Innovation

• Scaled mass transport with source-destination factors
• Models mass creation and destruction processes
• Establishes existence of optimal transport plans