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On the convergence of the generalized Weiszfeld algorithm. (English) Zbl 1176.90369
The paper deals with the convergence of the generalized Weiszfeld’s algorithm for solving of Weber-like problems. An objective of such a problem is to locate one service facility in the Euclidean space, in which n demands are located, so that the sum of specific functions depending on distances is minimal. Each of these functions depends on exactly one distance between the facility and one of the demand locations. In the original Weber problem, weighted Euclidean distances form the functions. The former Weiszfeld’s algorithm makes use of the recursive relationship derived from the stationary point condition. It was also given a proof of the algorithm convergence to a local minimum or saddle point. The generalized Weiszfeld’s algorithm based on the same recursive relation has been used to solve Weber-like problems unless the convergence has been proved. The author gives a thorough proof of the generalized Weiszfeld’s algorithm convergence together with alternative assumptions. Then he analyses necessity of the assumptions for the corollaries and derives a possible relaxation of the assumptions. In addition to this study, the author deals with the singular case, in which a local minimum of the problem lies at a demand point. The study is completed by an example and numerical results obtained from the solution of various instances. The paper is valuable for its instructiveness and insight into the proof methodology, which enable to other researchers to comply with similar problems of continuous location.
90B85Continuous location
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