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Weak optimal transport with moment constraints: constraintqualification, dual attainment and entropic regularization

Guillaume Carlier, Hugo Malamut, Maxime Sylvestre

TL;DR

This work extends weak optimal transport by incorporating both hard moment constraints and soft moment penalties, including martingale and vector-quantile regression settings. It develops a robust duality framework via Fenchel–Rockafellar theory, identifies a Slater-type constraint qualification that ensures dual attainment in unregularized and entropic cases, and analyzes the Gamma-convergence and convergence rates as entropy vanishes or penalties intensify. The authors connect MOT nondegeneracy and irreducibility to a $W_\infty$-stability criterion and implement a SISTA-based algorithm to compute solutions in dimension one, illustrating Brenier–Strassen and left-curtain interpolations. The results provide both theoretical guarantees for dual variables and practical schemes for computing constrained weak OT in finance and high-level regression contexts, along with detailed numerical experiments and interpolation schemes between canonical OT couplings.

Abstract

We consider weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglböck and Juillet.

Weak optimal transport with moment constraints: constraintqualification, dual attainment and entropic regularization

TL;DR

This work extends weak optimal transport by incorporating both hard moment constraints and soft moment penalties, including martingale and vector-quantile regression settings. It develops a robust duality framework via Fenchel–Rockafellar theory, identifies a Slater-type constraint qualification that ensures dual attainment in unregularized and entropic cases, and analyzes the Gamma-convergence and convergence rates as entropy vanishes or penalties intensify. The authors connect MOT nondegeneracy and irreducibility to a -stability criterion and implement a SISTA-based algorithm to compute solutions in dimension one, illustrating Brenier–Strassen and left-curtain interpolations. The results provide both theoretical guarantees for dual variables and practical schemes for computing constrained weak OT in finance and high-level regression contexts, along with detailed numerical experiments and interpolation schemes between canonical OT couplings.

Abstract

We consider weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglböck and Juillet.

Paper Structure

This paper contains 19 sections, 22 theorems, 187 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Moment constraints and qualification
  3. Martingale optimal transport
  4. Vector quantile regression
  5. Slater type condition and approximation with finite entropy
  6. Duality
  7. Setting
  8. A Fenchel-Rockafellar argument
  9. Relaxation
  10. Dual attainment
  11. The unregularized case
  12. The entropic case $\varepsilon>0$
  13. Primal-Dual optimality conditions
  14. Convergence of the primal problem
  15. Gamma-convergence
  16. ...and 4 more sections

Key Result

Lemma 2.3

Assume that $\mu \leq_{\mathrm{cvx}} \nu$ then where $C_\nu = \mathrm{co}(\mathrm{supp}(\nu))$ and $\mathrm{co}$ denotes the convex envelope.

Figures (2)

  • Figure 1: Interpolation between Brenier and Strassen.
  • Figure 2: Interpolation between the Left curtain transport and optimal transport

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2: BackhoffVeraguas2023ExistenceOB
  • Lemma 2.3
  • proof
  • Definition 2.4: backhoffveraguas2024gradientflowbassfunctional
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 41 more