  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 a modified Weiszfeld algorithm
presented by Vardi and Zhang in 2001.
  It can be seen that the proposed algorithm generates a sequence of feasible
iterates that have descent properties. Under certain hypotheses, the limit of
this sequence satisfies the KKT optimality conditions, is a fixed point of the
iteration function that defines the algorithm, and is the solution of the
constrained minimization problem. Numerical experiments confirmed the
theoretical results.
