35 research outputs found
A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization
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 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 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.Comment: 21 pages, 3 figure
A Trajectory Method for the Optimization of the Multi-Facility Location Problem With lp Distances
We consider the multi-facility location problem of placing m new facilities optimally among n demand points (or existing facilities) so that the sum of all weighted lp distance pairs (facility to facility and facility to demand point) is minimized. A method involving the solution of differential equations by numerical integration is presented. This method is computationally comparable to many existing heuristic and iterative methods. It avoids the frequent convergence difficulties associated with many iterative methods for p > 1 and has no difficulties in dealing with Targe "clusters" of facilities of p = 1.multifacility, location, lp, trajectory
A Nonlinear Approximation Method for Solving a Generalized Rectangular Distance Weber Problem
This paper provides a method for approximating optimal location in a multi-facility Weber problem where rectangular distances apply. Optimality is achieved when the sum of weighted distances is minimized. Two upper bounds on the error incurred by using the approximation are developed. The formulation can be used in convex programming to solve some nonlinearly constrained problems.