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A Weiszfeld-like algorithm for a Weber location problem constrained to a closed and convex set | Germán A. Torres
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5 Apr 2012 | Abstract: | The 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 | Services: | Forum | Review | PDF | Favorites |
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