In a recent issue of this journal, Mordukhovich, Nam, and Salinas pose and solve an interesting non-differentiable generalization of the Heron problem in the framework of modern convex analysis. In the generalized Heron problem, one is given k + 1 closed convex sets in ℝ d equipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the first k sets is minimal. In later work, the authors generalize the Heron problem even further, relax its convexity assumptions, study its theoretical properties, and pursue subgradient algorithms for solving the convex case. Here, we revisit the original problem solely from the numerical perspective. By exploiting the majorization-minimization (MM) principle of computational statistics and rudimentary techniques from differential calculus, we are able to construct a very fast algorithm for solving the Euclidean version of the generalized Heron problem.

中文翻译：

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从多数化-最小化的角度看广义 Heron 问题。

在本期刊的最近一期中，Mordukhovich、Nam 和 Salinas 在现代凸分析的框架内提出并解决了 Heron 问题的一个有趣的不可微概括。在广义 Heron 问题中，在 ℝ d 中给定 k + 1 个闭凸集，并配备其欧几里得范数，并要求在最后一个集中找到点，使得到前 k 个集的距离之和最小。在后来的工作中，作者进一步概括了 Heron 问题，放宽了它的凸性假设，研究了它的理论性质，并寻求解决凸情况的次梯度算法。在这里，我们仅从数值角度重新审视原始问题。通过利用计算统计的主要化-最小化 (MM) 原理和微分学的基本技术，