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

TL;DR

This paper studies weak optimal transport problems with moment constraints, establishing duality and convergence results, and demonstrates their application through numerical solutions of specific transport problems.

Contribution

It provides new duality and regularization results for weak optimal transport with moment constraints, including the martingale case, and offers numerical methods for these problems.

Findings

Established dual attainment under a qualification condition.

Analyzed convergence of entropic regularization schemes.

Numerically solved dimension-one transport problems, including the Brenier-Strassen problem.

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 Beiglb\"{o}ck and Juillet.