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29 March 2024
 
  » arxiv » 1204.1087

 Article overview


A Weiszfeld-like algorithm for a Weber location problem constrained to a closed and convex set
Germán A. Torres ;
Date 5 Apr 2012
AbstractThe Weber problem consists of finding a point in $mathbbm{R}^n$ that minimizes the weighted sum of distances from $m$ points in $mathbbm{R}^n$ that are not collinear. An application that motivated this problem is the optimal location of facilities in the 2-dimensional case. A classical method to solve the Weber problem, proposed by Weiszfeld in 1937, is based on a fixed point iteration.
In this work a Weber problem constrained to a closed and convex set is considered. A Weiszfeld-like algorithm, well defined even when an iterate is a vertex, is presented. The iteration function $Q$ that defines the proposed algorithm, is based mainly on an orthogonal projection over the feasible set, combined with the iteration function of the modified Weiszfeld algorithm presented by Vardi and Zhang in 2001. It can be proved that $x^*$ is a fixed point of the iteration function $Q$ if and only if $x^*$ is the solution of the constrained Weber problem. Besides that, under certain hypotheses, $x^*$ satisfies the KKT optimality conditions. The algorithm generates a sequence of feasible points having descending properties. The limit of this sequence is a fixed point of the iteration function $Q$, and therefore it is the solution of the constrained Weber problem. Numerical results confirmed the theoretical results and the robustness of the method.
Source arXiv, 1204.1087
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