2012/04/04 by G. Torres, Torres, Germán A.
Business, Management and Accounting · Computer Science · Mathematics · #90B85 #90C25 #90C30 #Computational Geometry and Mesh Generation #FOS: Mathematics #Facility Location and Emergency Management #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1204.1087
openalex publication_date 2012/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Weber problem consists of finding a point in mathbbmRn that\nminimizes the weighted sum of distances from m points in mathbbmRn that\nare not collinear. An application that motivated this problem is the optimal\nlocation of facilities in the 2-dimensional case. A classical method to solve\nthe Weber problem, proposed by Weiszfeld in 1937, is based on a fixed point\niteration.\n In this work a Weber problem constrained to a closed and convex set is\nconsidered. A Weiszfeld-like algorithm, well defined even when an iterate is a\nvertex, is presented. The iteration function Q that defines the proposed\nalgorithm, is based mainly on an orthogonal projection over the feasible set,\ncombined with the iteration function of a modified Weiszfeld algorithm\npresented by Vardi and Zhang in 2001.\n It can be seen that the proposed algorithm generates a sequence of feasible\niterates that have descent properties. Under certain hypotheses, the limit of\nthis sequence satisfies the KKT optimality conditions, is a fixed point of the\niteration function that defines the algorithm, and is the solution of the\nconstrained minimization problem. Numerical experiments confirmed the\ntheoretical results.\n