zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The Fermat-Weber location problem revisited. (English) Zbl 0855.90075
Summary: The Fermat-Weber location problem requires finding a point in N that minimizes the sum of weighted Euclidean distances to m given points. A one-point iterative method was first introduced by Weiszfeld in 1937 to solve this problem. Since then several research articles have been published on the method and generalizations thereof. Global convergence of Weiszfeld’s algorithm was proven in a seminal paper by Kuhn in 1973. However, since the m given points are singular points of the iteration functions, convergence is conditional on none of the iterates coinciding with one of the given points. In addressing this problem, Kuhn concluded that whenever the m given points are not collinear, Weiszfeld’s algorithm will converge to the unique optimal solution except for a denumerable set of starting points. As late as 1989, Chandrasekaran and Tamir demonstrated with counter-examples that convergence may not occur for continuous sets of starting points when the given points are contained in an affine subspace of N . We resolve this open question by proving that Weiszfeld’s algorithm converges to the unique optimal solution for all but a denumerable set of starting points if, and only if, the convex hull of the given points is of dimension N.

MSC:
90B85Continuous location
References:
[1]J. Brimberg and R.F. Love, ”Local convergence in a generalized Fermat–Weber problem,”Annals of Operations Research 40 (1992) 33–66. · Zbl 0787.90040 · doi:10.1007/BF02060469
[2]J. Brimberg and R.F. Love, ”Global convergence of a generalized iterative procedure for the minisum location problem withl p distances,”Operations Research 41 (1993) 1153–1163. · Zbl 0795.90037 · doi:10.1287/opre.41.6.1153
[3]R. Chandrasekaran and A. Tamir, ”Open questions concerning Weiszfeld’s algorithm for the Fermat–Weber location problem,”Mathematical Programming 44 (1989) 293–295. · Zbl 0683.90026 · doi:10.1007/BF01587094
[4]L. Cooper, ”Location–allocation problems,”Operations Research 11 (1963) 37–52. · Zbl 0113.14201 · doi:10.1287/opre.11.3.331
[5]H. Juel and R.F. Love, ”Fixed point optimality criteria for the location problem with arbitrary norms,”Journal of the Operational Research Society 32 (1981) 891–897.
[6]I.N. Katz, ”Local convergence in Fermat’s problem,”Mathematical Programming 6 (1974) 89–104. · Zbl 0291.90069 · doi:10.1007/BF01580224
[7]H.W. Kuhn, ”A note on Fermat’s problem,”Mathematical Programming 4 (1973) 98–107. · Zbl 0255.90063 · doi:10.1007/BF01584648
[8]H.W. Kuhn and R.E. Kuenne, ”An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics,”Journal of Regional Science 4 (1962) 21–34. · doi:10.1111/j.1467-9787.1962.tb00902.x
[9]W. Miehle, ”Link-length minimization in networks,”Operations Research 6 (1958) 232–243. · doi:10.1287/opre.6.2.232
[10]A.E. Taylor and W.R. Mann,Advanced Calculus (Xerox College Publishing, Lexington, MA, 2nd ed., 1972).
[11]E. Weiszfeld, ”Sur le point par lequel la somme des distances den points donnés est minimum,”Tohoku Mathematics Journal 43 (1937) 355–386.