title: A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization.
abstract: 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 givenk+ 1 closed convex sets in ℝdequipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the firstksets 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.
authors.affiliation: Department of Human Genetics, University of California, Los Angeles, CA 90095, ecchi@ucla.edu.
authors.forename: Eric C
authors.initials: EC
authors.lastname: Chi
authors.affiliation: Departments of Human Genetics, Biomathematics, and Statistics, University of California, Los Angeles, CA 90095, klange@ucla.edu.
authors.forename: Kenneth
authors.initials: K
authors.lastname: Lange