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The Fundamental Theorem of Weak Optimal Transport

Mathias Beiglböck, Gudmund Pammer, Lorenz Riess, Stefan Schrott

TL;DR

The paper develops a fundamental theorem for weak optimal transport (WOT), extending the classical OT structure to nonlinear, barycentric costs and establishing primal attainment, dual attainment, and complementary slackness via the C-transform. It unifies a range of results—convex Kantorovich–Rubinstein, Brenier–Strassen, Gangbo–McCann–Strassen, entropic OT, and martingale transport—under a single duality framework and shows how to handle nonconvex costs through relaxed formulations. The authors derive detailed optimality conditions, C-monotonicity, and regularization schemes (entropic and convex) that yield explicit Gibbs-type structure in the optimal couplings. They further connect these results to financial mathematics via convex order, and demonstrate how relaxed WMOT yields tractable models for martingale transport problems such as the martingale Benamou–Brenier and entropic martingale transport. Overall, the work broadens OT’s applicability to nonlinear costs, offers unified proofs, and provides tools for robust and regularized transport in finance and related fields.

Abstract

The fundamental theorem of classical optimal transport establishes strong duality and characterizes optimizers through a complementary slackness condition. Milestones such as Brenier's theorem and the Kantorovich-Rubinstein formula are direct consequences. In this paper, we generalize this result to non-linear cost functions, thereby establishing a fundamental theorem for the weak optimal transport problem introduced by Gozlan, Roberto, Samson, and Tetali. As applications we provide concise derivations of the Brenier--Strassen theorem, the convex Kantorovich--Rubinstein formula and the structure theorem of entropic optimal transport. We also extend Strassen's theorem in the direction of Gangbo--McCann's transport problem for convex costs. Moreover, we determine the optimizers for a new family of transport problems which contains the Brenier--Strassen, the martingale Benamou--Brenier and the entropic martingale transport problem as extreme cases.

The Fundamental Theorem of Weak Optimal Transport

TL;DR

The paper develops a fundamental theorem for weak optimal transport (WOT), extending the classical OT structure to nonlinear, barycentric costs and establishing primal attainment, dual attainment, and complementary slackness via the C-transform. It unifies a range of results—convex Kantorovich–Rubinstein, Brenier–Strassen, Gangbo–McCann–Strassen, entropic OT, and martingale transport—under a single duality framework and shows how to handle nonconvex costs through relaxed formulations. The authors derive detailed optimality conditions, C-monotonicity, and regularization schemes (entropic and convex) that yield explicit Gibbs-type structure in the optimal couplings. They further connect these results to financial mathematics via convex order, and demonstrate how relaxed WMOT yields tractable models for martingale transport problems such as the martingale Benamou–Brenier and entropic martingale transport. Overall, the work broadens OT’s applicability to nonlinear costs, offers unified proofs, and provides tools for robust and regularized transport in finance and related fields.

Abstract

The fundamental theorem of classical optimal transport establishes strong duality and characterizes optimizers through a complementary slackness condition. Milestones such as Brenier's theorem and the Kantorovich-Rubinstein formula are direct consequences. In this paper, we generalize this result to non-linear cost functions, thereby establishing a fundamental theorem for the weak optimal transport problem introduced by Gozlan, Roberto, Samson, and Tetali. As applications we provide concise derivations of the Brenier--Strassen theorem, the convex Kantorovich--Rubinstein formula and the structure theorem of entropic optimal transport. We also extend Strassen's theorem in the direction of Gangbo--McCann's transport problem for convex costs. Moreover, we determine the optimizers for a new family of transport problems which contains the Brenier--Strassen, the martingale Benamou--Brenier and the entropic martingale transport problem as extreme cases.

Paper Structure

This paper contains 31 sections, 31 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Applications of the Fundamental Theorem of WOT
  3. Convex Kantorovich--Rubinstein
  4. The Brenier--Strassen theorem
  5. A Gangbo--McCann--Strassen theorem
  6. Entropic optimal transport
  7. Relaxation of martingale transport
  8. From Brenier--Strassen to martingale Benamou--Brenier
  9. Relaxing martingale EOT
  10. Duality and dual attainment without convexity assumptions
  11. Related literature
  12. Fundamental theorem of weak optimal transport
  13. Primal attainment and duality
  14. Complementary slackness
  15. Dual attainment
  16. ...and 16 more sections

Key Result

Theorem 1.1

For $c:X\times Y\to [0,\infty]$ lsc, the primal problem ${\rm T}_c(\mu, \nu)$ is attained and equals the dual problem The dual problem is attained if $c$ is upper bounded in the sense that Candidates $\pi, (f, g)$ are optimal if and only if they satisfy the complementary slackness condition

Theorems & Definitions (76)

  • Theorem 1.1: Fundamental Theorem of OT
  • Theorem 1.2: Fundamental Theorem of WOT
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 66 more