On the Convergence of a Class of Iterative Methods for Solving the Weber Location Problem

The location problem is to find a point M whose sum of weighted distances from m vertices in p-dimensional Euclidean space is a minimum. The best-known algorithm for solving the location problem is an iterative scheme devised by Weiszfeld in 1937. The procedure will not converge if some nonoptimal vertex is an iterate, however. This paper solves the problem of vertex iterates and presents a general proof permitting a variable step length within certain bounds. This property is used, in particular, to show the convergence of a modified gradient Newton-Raphson type of procedure.

[1]  W. C. Eells.  A Mistaken Conception of the Center of Population , 1930 .

[2]  R. Courant,et al.  What Is Mathematics , 1943 .

[3]  O. Lindberg.  An Economic-Geographical Study of the Localization of the Swedish Paper Industry , 1953 .

[4]  J. Hart.  Central Tendency in Areal Distributions , 1954 .

[5]  W. Miehle.  Link-Length Minimization in Networks , 1958 .

[6]  P. Porter.  WHAT IS THE POINT OF MINIMUM AGGREGATE TRAVEL?1 , 1963 .

[7]  L. Cooper.  Location-Allocation Problems , 1963 .

[8]  A. Court.  THE ELUSIVE POINT OF MINIMUM TRAVEL , 1964 .

[9]  P. Henrici.  Elements of numerical analysis , 1966 .

[10]  R. Bellman.  An Application of Dynamic Programming to Location—Allocation Problems , 1965 .

[11]  Samuel Eilon,et al.  Siting a Distribution Centre, An Analogue Computer Application , 1966 .

[12]  H. Keller,et al.  Analysis of Numerical Methods , 1969 .

[13]  W. Alonso.  A REFORMULATION OF CLASSICAL LOCATION THEORY AND ITS RELATION TO RENT THEORY , 1967 .

[14]  Leon Cooper,et al.  AN EXTENSION OF THE GENERALIZED WEBER PROBLEM , 1968 .

[15]  I. Katz.  On the Convergence of a Numerical Scheme for Solving Some Locational Equilibrium Problems , 1969 .

[16]  D. H. Marks,et al.  An Analysis of Private and Public Sector Location Models , 1970 .

[17]  Gunnar Törnqvist,et al.  Multiple location analysis , 1971 .

[18]  Harold W. Kuhn,et al.  A note on Fermat's problem , 1973, Math. Program..

[19]  I. Norman Katz,et al.  Local convergence in Fermat's problem , 1974, Math. Program..

[20]  A. P. Hurter,et al.  An Efficient Algorithm for the Solution of the Weber Problem with Mixed Norms , 1975 .