Dynamic characterization of barycentric optimal transport problems and their martingale relaxation

2025-11-26

AI Summary

This study establishes a dynamic characterization of barycentric optimal transport within the weak optimal transport framework. To this end, we first extend the Benamou–Brenier formula to a weak formulation for barycentric transport, yielding a dynamic variational representation. We then introduce a martingale relaxation structure and systematically establish its equivalence and nested relationship with the martingale Benamou–Brenier formula recently proposed by Backhoff-Veraguas et al. Methodologically, our approach integrates weak optimal transport theory, convex analysis, stochastic processes, and dynamic programming principles, employing martingale-constrained optimization for modeling and analysis. The main contributions are: (i) establishing a canonical dynamic representation paradigm for barycentric transport; (ii) bridging weak transport and martingale transport theories; and (iii) providing a rigorous foundation for numerical algorithms and probabilistic applications in this setting.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We extend the Benamou-Brenier formula from classical optimal transport to weak optimal transport and show that the barycentric optimal transport problem studied by Gozlan and Juillet has a dynamic analogue. We also investigate a martingale relaxation of this problem, and relate it to the martingale Benamou-Brenier formula of Backhoff-Veraguas, Beiglböck, Huesmann and Källblad.

Problem

• Extend Benamou-Brenier formula to weak optimal transport
• Develop dynamic analogue for barycentric optimal transport problems
• Investigate martingale relaxation of barycentric transport problems

Innovation

• Extended Benamou-Brenier formula to weak transport
• Developed dynamic analogue for barycentric optimal transport
• Investigated martingale relaxation with related formulas