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A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem. (English) Zbl 1189.90086
Summary: The Generalized Fermat Problem (in the plane) is: given n3 destination points find the point x * which minimizes the sum of Euclidean distances from x * to each of the destination points. The Weiszfeld iterative algorithm for this problem is globally convergent, independent of the initial guess. Also, a test is available, a priori, to determine when x * is a destination point. This paper generalizes earlier work by the first author by introducing an asymmetric Euclidean distance in which, at each destination, the x-component is weighted differently from the y-component. A Weiszfeld algorithm is studied to compute x * and is shown to be a descent method which is globally convergent (except possibly for a denumerable number of starting points). Local convergence properties are characterized. When x * is not a destination point, the iteration matrix at x * is shown to be convergent and local convergence is always linear. When x * is a destination point, local convergence can be linear, sub-linear or super-linear, depending upon a computable criterion. A test, which does not require iteration, for x * to be a destination, is derived. Comparisons are made between the symmetric and asymmetric problems. Numerical examples are given.
MSC:
90B85Continuous location
52B55Computational aspects related to geometric convexity
References:
[1]Cooper, L.: Location–allocation problems, Oper. res. 11, 331-343 (1963) · Zbl 0113.14201 · doi:10.1287/opre.11.3.331
[2]Cooper, L.: Heuristic methods for location–allocation problems, SIAM rev. 6, 37-52 (1964)
[3]Cooper, L.: Solutions of generalized locational equilibrium problems, J. regional sci. 7, 1-18 (1967)
[4]Cooper, L.: An extension of the generalized Weber problem, J. regional sci. 8, 181-197 (1968)
[5]Cooper, L.; Katz, I. Norman: The Weber problem revisited, J. comput. Math. appl. 7, 225-235 (1981) · Zbl 0457.65044 · doi:10.1016/0898-1221(81)90082-1
[6]Katz, I. Norman: On the convergence of a numerical scheme for solving some locational equilibrium problems, SIAM J. Appl. math. 17, 1224-1231 (1969) · Zbl 0187.18001 · doi:10.1137/0117113
[7]Katz, I. Norman: Local convergence in Fermat’s problem, Math. program. 6, 89-104 (1974) · Zbl 0291.90069 · doi:10.1007/BF01580224
[8]Kuhn, H. W.; Kuenne, R. E.: An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics, J. regional sci. 4, 21-23 (1962)
[9]Kuhn, H. W.: A note on Fermat’s problem, Math. program. 4, 98-107 (1973) · Zbl 0255.90063 · doi:10.1007/BF01584648
[10]Miehle, W.: Link-length minimization in networks, Oper. res. 6, 232-243 (1958)
[11]Plastria, Frank; Elosmani, Mohamed: On the convergence of the weiszfeld algorithm for continuous single facility location–allocation problems, Top 16, 388-406 (2008) · Zbl 1154.90531 · doi:10.1007/s11750-008-0056-1
[12]Morris, James G.: Convergence of the weiszfeld algorithm for Weber problems using a generalized ”distance” function, Oper. res. 29, 37-47 (1981) · Zbl 0452.90023 · doi:10.1287/opre.29.1.37
[13]Xue, G.; Ye, Y.: An efficient algorithm for minimizing a sum of P-norms, SIAM J. Optim. (1997)
[14]Eckhardt, Ulrich: Weber’s problem and weiszfeld’s algorithm in general spaces, Math. program. 18, 186-196 (1980) · Zbl 0433.65035 · doi:10.1007/BF01588313
[15]Li, Y.: A Newton acceleration of the weiszfeld algorithm for minimizing the sum of Euclidean distances, Comput. optim. Appl. 10, 219-242 (1998) · Zbl 0912.90197 · doi:10.1023/A:1018333422414
[16]Edmund, K. A.; Christiansen, E.; Conn, A. R.; Overton, M. L.: An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms, SIAM J. Sci. comput. (1998)
[17]Weiszfeld, E.: Sur le point pour lequel la sommme des distances de n points donnes est minimum, Tohoku math. J. 43, 355-386 (1937) · Zbl 0017.18007 · doi:10.2748/tmj/1178227459
[18]Golub, G. H.; Van Loan, C. F.: Matrix computations, (1989)