We propose a numerical algorithm for the computation of multi-marginal optimal transport (MMOT) problems involving general probability measures that are not necessarily discrete. By developing a relaxation scheme in which marginal constraints are replaced by finitely many linear constraints and by proving a specifically tailored duality result for this setting, we approximate the MMOT problem by a linear semi-infinite optimization problem. Moreover, we are able to recover a feasible and approximately optimal solution of the MMOT problem, and its sub-optimality can be controlled to be arbitrarily close to 0 under mild conditions. The developed relaxation scheme leads to a numerical algorithm which can compute a feasible approximate optimizer of the MMOT problem whose theoretical sub-optimality can be chosen to be arbitrarily small. Besides the approximate optimizer, the algorithm is also able to compute both an upper bound and a lower bound for the optimal value of the MMOT problem. The difference between the computed bounds provides an explicit sub-optimality bound for the computed approximate optimizer. We demonstrate the proposed algorithm in three numerical experiments involving an MMOT problem that stems from fluid dynamics, the Wasserstein barycenter problem, and a large-scale MMOT problem with 100 marginals. We observe that our algorithm is capable of computing high-quality solutions of these MMOT problems and the computed sub-optimality bounds are much less conservative than their theoretical upper bounds in all the experiments.

翻译：本文提出了一种用于计算涉及一般概率测度（不一定是离散测度）的多边际最优传输（MMOT）问题的数值算法。通过构建一种松弛方案——其中边际约束被有限多个线性约束所替代，并为此设定证明了一个专门定制的对偶性结果，我们将MMOT问题近似为一个线性半无限优化问题。此外，我们能够恢复MMOT问题的一个可行且近似最优的解，并且在温和条件下，其次优性可以被控制到任意接近0。所开发的松弛方案导出了一个数值算法，该算法能够计算MMOT问题的一个可行近似优化器，其理论次优性可以选择为任意小。除了近似优化器外，该算法还能够计算MMOT问题最优值的上界和下界。计算所得上下界之间的差值，为计算出的近似优化器提供了一个明确的次优性界限。我们在三个数值实验中演示了所提出的算法，这些实验涉及一个源自流体动力学的MMOT问题、Wasserstein重心问题，以及一个具有100个边际的大规模MMOT问题。我们观察到，我们的算法能够计算这些MMOT问题的高质量解，并且在所有实验中，计算得到的次优性界限远小于其理论上界，保守性大大降低。