A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization

TL;DR

This paper introduces a fast algorithm for solving the generalized Heron problem in Euclidean space by applying the majorization-minimization principle, focusing on numerical solutions rather than theoretical analysis.

Contribution

The paper develops a novel, efficient numerical algorithm for the Euclidean generalized Heron problem using the majorization-minimization approach.

Findings

The algorithm converges rapidly to the optimal solution.

It outperforms existing methods in computational speed.

The approach is simple to implement and effective.

Abstract

In a recent issue of this journal, Mordukhovich et al.\ 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 $\Real^d$ 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.