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We propose a numerical algorithm for the computation of multi-marginaloptimal transport (MMOT) problems involving general measures that are notnecessarily discrete. By developing a relaxation scheme in which marginalconstraints are replaced by finitely many linear constraints and by proving aspecifically tailored duality result for this setting, we approximate the MMOTproblem by a linear semi-infinite optimization problem. Moreover, we are ableto recover a feasible and approximately optimal solution of the MMOT problem,and its sub-optimality can be controlled to be arbitrarily close to 0 undermild conditions. The developed relaxation scheme leads to a numerical algorithmwhich can compute a feasible approximate optimizer of the MMOT problem whosetheoretical sub-optimality can be chosen to be arbitrarily small. Besides theapproximate optimizer, the algorithm is also able to compute both an upperbound and a lower bound on the optimal value of the MMOT problem. Thedifference between the computed bounds provides an explicit upper bound on thesub-optimality of the computed approximate optimizer. Through a numericalexample, we demonstrate that the proposed algorithm is capable of computing ahigh-quality solution of an MMOT problem involving as many as 50 marginalsalong with an explicit estimate of its sub-optimality that is much lessconservative compared to the theoretical estimate.

Numerical method for feasible and approximately optimal solutions of multi-marginal optimal transport beyond discrete measures