Semi-discrete optimal transport setting is a very important formulation in the computation of Wasserstein distance, as it is an approximation form of the continuous setting of optimal transport. However, initialization process of dual weight vector for the dual problem in this setting is required for the computation of the first and second order methods, since the Laguerre cells associated to the dual weight vector ought to have positive mass during the calculation. Although there are approaches to initialize the weight vector, systematic computing procedure has not been available. To resolve this problem, we discretize the domain of source distribution of this problem, approximate the Laguerre cells with respect to the weight vector, and finally employ the local perturbation method (Jocelyn, 2019) combined with boundary method to force all measures of the Laguerre cells great than zero. In addition, we study the corresponding theoretical results of the computation process and provide the algorithm, ensuring that the problem can indeed be efficiently computed.

半离散最优运输设置是Wasserstein距离计算中非常重要的公式，因为它是最优运输连续设置的近似形式。但是，在这种情况下，一阶和二阶方法的计算需要针对双重问题的双重权重向量的初始化过程，因为与双重权重向量关联的Laguerre像元在计算过程中应具有正质量。尽管有初始化权重向量的方法，但是系统的计算过程还不可用。为了解决这个问题，我们离散化了该问题的源分布域，根据权重向量对Laguerre单元进行了近似，最后采用了局部摄动法（Jocelyn，2019）结合边界法将Laguerre像元的所有量度强制大于零。另外，我们研究了计算过程的相应理论结果并提供了算法，以确保确实可以有效地解决问题。