In this work, we propose a novel implementation of Newton's method for solving semi-discrete optimal transport (OT) problems for cost functions which are a positive combination of $p$-norms, $1<p<\infty$. It is well understood that the solution of a semi-discrete OT problem is equivalent to finding a partition of a bounded region in Laguerre cells, and we prove that the Laguerre cells are star-shaped with respect to the target points. By exploiting the geometry of the Laguerre cells, we obtain an efficient and reliable implementation of Newton's method to find the sought network structure. We provide implementation details and extensive results in support of our technique in 2-d problems, as well as comparison with other approaches used in the literature.

翻译：本文提出了一种新的牛顿法实现，用于求解成本函数为 $p$-范数正组合（$1<p<\infty$）的半离散最优输运（OT）问题。众所周知，半离散OT问题的解等价于在有界区域内划分拉盖尔胞腔，我们证明了这些拉盖尔胞腔相对于目标点具有星形性质。通过利用拉盖尔胞腔的几何结构，我们获得了一种高效可靠的牛顿法实现以寻找所需的网络结构。我们提供了二维问题的实现细节和大量实验结果，并与文献中其他方法进行了比较。