Douglas–Rachford Splitting for High-Accuracy Unregularised Multi-Marginal Optimal Transport

Multi-Marginal Optimal Transport (MMOT) is commonly solved with entropic regularization because iterative scaling is fast and stable at moderate accuracy. The same regularization, however, introduces a non-vanishing bias with respect to the original linear program. We study DRMOT, a Douglas–Rachford splitting method for unregularised MMOT. Its key technical ingredient is a closed-form orthogonal projection onto the affine set of prescribed tensor marginals, which extends the known matrix projection to arbitrary-order tensors. We further introduce a scale-aware default step size and a cost-aware feasible recovery tailored to the sparse iterates produced by DR; on identical DRMOT iterates, this recovery reduces the final reported error by roughly $5$–$10\times$ relative to the standard rank-one correction. We re-evaluate DRMOT under equal wall-clock budgets as a complete finite-time solver: the entropic baselines retain their published recovery rules, while DRMOT uses the proposed cost-aware recovery. On Wasserstein-barycenter grids, the best DRMOT variant is $2.1\times$–$15.7\times$ more accurate than the best tuned entropic baseline within the same 20-second budget. In within-family cost diagnostics over the two evaluated instances per family, the best observed DRMOT-s runs are the only tested ones to reach an absolute objective gap of $10^{-4}$ within 20 seconds on both Coulomb and Brenier–Euler costs, while the dimension-only DRMOT variant gives the best observed $10^{-3}$-level time on risk pooling. On model-independent option pricing, DRMOT agrees with a high-accuracy HiGHS reference down to float64 numerical precision whereas the entropic methods plateau near $2\times 10^{-4}$. These results identify the regime in which solving MMOT without entropic bias is practically beneficial.