The extended Weber location problem is a classical optimization problem that has inspired some new works in several machine learning scenarios recently. However, most existing algorithms may get stuck due to the singularity at the data points when the power of the cost function $1\leqslant q<2$, such as the widely-used iterative Weiszfeld approach. In this paper, we establish a de-singularity subgradient approach for this problem. We also provide a complete proof of convergence which has fixed some incomplete statements of the proofs for some previous Weiszfeld algorithms. Moreover, we deduce a new theoretical result of superlinear convergence for the iteration sequence in a special case where the minimum point is a singular point. We conduct extensive experiments in a real-world machine learning scenario to show that the proposed approach solves the singularity problem, produces the same results as in the non-singularity cases, and shows a reasonable rate of linear convergence. The results also indicate that the $q$-th power case ($1<q<2$) is more advantageous than the $1$-st power case and the $2$-nd power case in some situations. Hence the de-singularity subgradient approach is beneficial to advancing both theory and practice for the extended Weber location problem.

在这篇论文中，我们针对扩展的韦伯位置问题，建立了一种去奇性亚梯度方法，并提供了完整的收敛证明。我们还推导了在极小点为奇异点的特殊情况下迭代序列的超线性收敛性质。实验结果表明，所提出的方法解决了奇性问题，在非奇性情况下产生了相同的结果，并展示了合理的线性收敛速度。

对于延伸的韦伯选址问题的去奇异子梯度方法