# A globally convergent algorithm for the Euclidean multiplicity location problem

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## Abstract

The Euclidean single facility location problem (ESFL) and the Euclidean multiplicity location problem (EMFL) are two special nonsmooth convex programming problems which have attracted a large literature. For the ESFL problem, there are algorithms which converge both globally and quadratically. For the EMFL problem, there are some quadratically convergent algorithms, but for global convergence, they all need nontrivial assumptions on the problem.

In this paper, we present an algorithm for EMFL. With no assumption on the problem, it is proved that from any initial point, this algorithm generates a sequence of points which converges to the closed convex set of optimal solutions of EMFL.

### Keywords

Programming Problem Initial Point Location Problem Facility Location Global Convergence## Preview

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### References

- [1]Armijo, L., Minimization of Functions having Lipschitz Continuous First Partial Derivatives,
*Pacific Journal of Mathematics*,**10**(1966), 1–3.Google Scholar - [2]Calamai, P.H. and Conn, A.R., A Stable Algorithm for Solving the Multifacility Location Problem involving Euclidean Distances,
*SIAM Journal on Scientific and Statistical Computing*,**1**(1980), 512–525Google Scholar - [3]Calamai, P.H. and Conn, A.R., A Second-Order Method for Solving the Continuous Multifacility Location Problem, in Watson, G.A. ed., Numerical Analysis: Proceedings of the Ninth Biennial Conference, Dundee, Scotland, Lecture Notes in Mathematics 912, Springer-Verlag, 1982, 1–25.Google Scholar
- [4]Calamai, P.H. and Conn, A.R., A Projected Newton Method for
*l*_{p}Norm Location Problems,*Mathematical Programming*,**38**(1987), 75–109.Google Scholar - [5]Dax, A., A Note on Optimality Conditions for the Euclidean Multifacility Location Problem,
*Mathematical Programming*,**36**(1986), 72–80.Google Scholar - [6]Fletcher, R., Practical Methods of Optimization, Second Edition, John Wiley & Sons, 1987.Google Scholar
- [7]Francis, R.L. and Cabot, A.V., Properties of a Multifacility Location Problem involving Euclidean Distances,
*Naval Research Logistics Quarterly*,**19**(1972), 335–353.Google Scholar - [8]Francis, R.L. and Goldstein, J.M., Location Theory: A Selective Bibliography,
*Operations Research*,**22**(1974), 400–410.Google Scholar - [9]
- [10]Levitin, E.S. and Polyak, D.T., Constrained Minimization Methods,
*USSR Computational Mathematics and Mathematical Physics*,**6**:5 (1966), 1–50.Google Scholar - [11]Love, R.F., Morris, J.G. and Wesolowsky, G.O., Facilities Location: Models & Methods, North-Holland, 1988.Google Scholar
- [12]Miehle, W., Link Length Minimization in Networks,
*Operations Research*,**6**(1958), 232–243.Google Scholar - [13]Morris, J.G., Convergence of the Weiszfeld Algorithm for the Weber Problem using a Generalized “Distance” Function,
*Operations Research*,**29**(1981), 37–48.Google Scholar - [14]Ostresh, L.M., The Multifacility Location Problem: Applications and Descent Theorems,
*Journal of Regional Science*,**17**(1977), 409–419.Google Scholar - [15]Ostresh, L.M., On the Convergence of a Class of Iterative Methods for Solving the Weber Location Problem,
*Operations Research*,**26**(1978), 597–609.Google Scholar - [16]Overton, M.L., A Quadratically Convergent Method for Minimizing a Sum of Euclidean Norms,
*Mathematical Programming*,**27**(1983), 34–63.Google Scholar - [17]Rado, F., The Euclidean Multifacility Location Problem,
*Operations Research*,**36**(1988), 485–492.Google Scholar - [18]Rockafellar, R.T., Convex Analysis, Princeton University Press, 1970.Google Scholar
- [19]Rosen, J.B. and Xue, G.L., On the Convergence of Miehle's Algorithm for the Euclidean Multifacility Location Problem, TR 90-27, Computer Science Department, University of Minnesota,
*Operations Research*, to appear.Google Scholar - [20]Wang, C.Y. et al, On the Convergence and Rate of Convergence of an Iterative Algorithm for the Plant Location Problem,
*Qufu Shiyun Xuebao*,**2**(1975), 14–25.Google Scholar - [21]Wang, C.Y., Minimizing Σ
_{i=1}^{m}*c*_{i}∥*x*−*a*_{i}∥ on a Closed Convex Set,*Acta Mathematicae Applicatae Sinica*,**1**(1978), 145–150.Google Scholar - [22]Wang, C.Y., On a New Pivot Method and the Convergence of the Levitin-Polayk Gradient Projection Method,
*Acta Mathematicae Applicatae Sinica*,**4**(1981), 37–52.Google Scholar - [23]Weiszfeld, E., Sur le Point par Lequel le Somme des Distances de
*n*Points donnes est Mininmum,*Tohoku Mathematical Journal*,**43**(1937), 355–386.Google Scholar - [24]Xue, G.L., A Globally and Quadratically Convergent Algorithm for min Σ
_{i=1}^{m}*c*_{i}∥*x*−*a*_{i}∥ Type Plant Location Problem,*Acta Mathematicae Applicatae Sinica*,**12**(1989), 65–72.Google Scholar

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© Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A. 1992