We propose a numerical algorithm for the computation of multi-marginal optimal transport (MMOT) problems involving general 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 on the optimal value of the MMOT problem. The difference between the computed bounds provides an explicit upper bound on the sub-optimality of the computed approximate optimizer. Through a numerical example, we demonstrate that the proposed algorithm is capable of computing a high-quality solution of an MMOT problem involving as many as 50 marginals along with an explicit estimate of its sub-optimality that is much less conservative compared to the theoretical estimate.

翻译：本文提出了一种针对涉及不一定离散的一般测度的多边缘最优输运（MMOT）问题的数值算法。通过开发一种松弛方案，将边缘约束替换为有限多个线性约束，并针对该设定证明了一个特定的对偶结果，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问题的可行且近似最优解，且在温和条件下其次优性可被控制到任意接近0。所发展的松弛方案催生了一种数值算法，该算法能够计算MMOT问题的可行近似最优解，其理论次优性可选择为任意小。除近似最优解外，该算法还能同时计算MMOT问题最优值的上界和下界。计算所得边界之差为计算出的近似最优解的次优性提供了一个显式上界。通过一个数值示例，我们证明了所提算法能够计算包含多达50个边缘的MMOT问题的高质量解，并附带一个显式的次优性估计，该估计比理论估计保守得多。