A Unified Kantorovich Duality for Multimarginal Optimal Transport

2026-01-23

AI Summary

This work addresses the long-standing absence of a unified Kantorovich duality theory and existence guarantees for dual optimizers in multi-marginal optimal transport (MOT) over general Polish spaces, particularly in non-compact settings. By reformulating the dual problem through convex analysis, the authors extend the classical two-marginal c-conjugacy to a genuine multi-marginal framework, introducing the notions of c-splitting sets and multi-marginal c-cyclical monotonicity. A truncation-compactness argument is employed to handle non-compactness. On arbitrary product Polish spaces, they establish a complete primal–dual equality, prove the existence of dual optimizers, and provide a canonical c-conjugate representation for optimal dual potentials. These results lay a rigorous theoretical foundation for the stability, differentiability, and asymptotic analysis of MOT problems.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Multimarginal optimal transport (MOT) has gained increasing attention in recent years, notably due to its relevance in machine learning and statistics, where one seeks to jointly compare and align multiple probability distributions. This paper presents a unified and complete Kantorovich duality theory for MOT problem on general Polish product spaces with bounded continuous cost function. For marginal compact spaces, the duality identity is derived through a convex-analytic reformulation, that identifies the dual problem as a Fenchel-Rockafellar conjugate. We obtain dual attainment and show that optimal potentials may always be chosen in the class of $c$-conjugate families, thereby extending classical two-marginal conjugacy principle into a genuinely multimarginal setting. In non-compact setting, where direct compactness arguments are unavailable, we recover duality via a truncation-tightness procedure based on weak compactness of multimarginal transference plans and boundedness of the cost. We prove that the dual value is preserved under restriction to compact subsets and that admissible dual families can be regularized into uniformly bounded $c$-conjugate potentials. The argument relies on a refined use of $c$-splitting sets and their equivalence with multimarginal $c$-cyclical monotonicity. We then obtain dual attainment and exact primal-dual equality for MOT on arbitrary Polish spaces, together with a canonical representation of optimal dual potentials by $c$-conjugacy. These results provide a structural foundation for further developments in probabilistic and statistical analysis of MOT, including stability, differentiability, and asymptotic theory under marginal perturbations.

Problem

• Multimarginal Optimal Transport
• Kantorovich Duality
• Polish Spaces
• c-conjugacy
• Duality Attainment

Innovation

• multimarginal optimal transport
• Kantorovich duality
• c-conjugacy
• Fenchel-Rockafellar conjugate
• c-cyclical monotonicity