Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

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

Eric C. Chi and Kenneth Lange
The American Mathematical Monthly
Vol. 121, No. 2 (February 2014), pp. 95-108
DOI: 10.4169/amer.math.monthly.121.02.095
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.121.02.095
Page Count: 14
  • Download ($19.00)
  • Cite this Item
Item Type
Article
References
A Look at the Generalized Heron Problem
through the Lens of Majorization-Minimization
Preview not available

Abstract

Abstract In a recent issue of this Monthly, 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, we are given k + 1 closed convex sets in ℝd 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 majorizationminimization (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.

Page Thumbnails