article

A Stable Algorithm for Solving the Multifacility Location Problem Involving Euclidean Distances

Authors:

A. R. ConnAuthors Info & Claims

SIAM Journal on Scientific and Statistical Computing, Volume 1, Issue 4

Pages 512 - 526

https://doi.org/10.1137/0901037

Published: 01 December 1980 Publication History

Abstract

There is a rapidly growing interdisciplinary interest in the application of location models to real life problems. Unfortunately, the current methods used to solve the most popular minisum and minimax location problems are computationally inadequate. A more unified and numerically stable approach for solving these problems is proposed. Detailed analysis is done for the linearly constrained Euclidean distance minisum problem for facilities located in a plane. Preliminary computational experience suggests that this approach compares favourably with other methods.

References

[1]

R.H. Bartels and A. R. Conn, Linearly constrained discrete l₁ problems, ACM TOMS, 6 (1980), pp. 594-608.

[2]

R.H. Bartels, A. R. Conn, and J. W. Sinclair, Minimization techniques for piecewise differentiable functions: The l₁ solution to an overdetermined linear system, SIAM J. Numer. Anal., 15 (1978), pp. 224-241.

[3]

P.H. Calamai, and C. Charalambous, Solving multifacility location problems involving Euclidean distances, Naval Res. Log. Quart., to appear.

[4]

C. Charalambous and A. R. Conn, An efficient method to solve the minimax problem directly, SIAM J. Numer. Anal., 15 (1978), pp. 162-187.

[5]

J. A. Chatelon, D. W. Hearn, and T. J. Lowe, A subgradient algorithm for certain minimax and minisum problems, Math. Prog., 15 (1978), pp. 130-145.

[6]

T. F. Coleman and A. R. Conn, Nonlinear programming via an exact penalty function method: A global method, Comp. Sci. Tech. Rep. CS-80-31, University of Waterloo, Waterloo, Ontario, 1980 (to appear in Math. Prog.).

[7]

A. R. Conn, Constrained optimization using a nondifferentiable penalty function, SIAM J. Numer. Anal., 10 (1973), pp. 760-784.

[8]

A. R. Conn, and T. Pietrzykowski, A penalty function method converging directly to a constrained optimum, SIAM J. Numer. Anal., 14 (1977), pp. 348-375.

[9]

J. W. Eyster, J. A. White, and W. W. Wierwille, On solving multifacility location problems using a hyperboloid approximation procedure, AIIE Trans., 5 (1973), pp. 1-6.

[10]

R. L. Francis and A. V. Cabot, Properties of a multifacility location problem involving Euclidean distances, Naval Res. Log. Quart., 19 (1972), pp. 335-353.

[11]

R. L. Francis and J. M. Goldstein, Location theory: A selective bibliography, Oper. Res., 22 (1974), pp. 400-410.

[12]

P. E. Gill, G. H. Golub, W. Murray and M. Saunders, Methods for modifying matrix factorizations, Math. Comp., 28 (1974), pp. 505-535.

[13]

R. F. Love, Locating facilities in three-dimensional space by convex programming, Naval Res. Log. Quart., 16 (1969), pp. 503-516.

[14]

W. Murray and M. Overton, Steplength algorithms for minimizing a class of non-differentiable functions, Computing, 23 (1979), pp. 309-331.

[15]

T. Pietrzykowski, An exact potential method for constrained maxima, SIAM J. Numer. Anal., 6 (1969), pp. 299-304.

[16]

R. C. Vergin and J. D. Rogers, An algorithm and computational procedure for locating economic facilities, Management Sci., 13 (1967), pp. 240-254.

[17]

G. O. Wesolowsky and W. G. Truscott, Dynamic location of multiple facilities for a class of continuous problems, AIIE Trans., 8 (1976), pp. 76-83.

[18]

W. I. Zangwill, Nonlinear programming via penalty functions, Management Si., 13 (1962), pp. 344-350.

Cited By

He SJiang YPan S(2010)An entropy regularization technique for minimizing a sum of Tchebycheff normsApplied Numerical Mathematics10.1016/j.apnum.2009.11.00560:3(208-223)Online publication date: 1-Mar-2010
https://dl.acm.org/doi/10.1016/j.apnum.2009.11.005
Pan SJiang Y(2008)Smoothing Newton Method for Minimizing the Sum of p-NormsJournal of Optimization Theory and Applications10.1007/s10957-008-9364-8137:2(255-275)Online publication date: 1-May-2008
https://dl.acm.org/doi/10.1007/s10957-008-9364-8
Andersen KChristiansen EConn AOverton M(2000)An Efficient Primal-Dual Interior-Point Method for Minimizing a Sum of Euclidean NormsSIAM Journal on Scientific Computing10.1137/S106482759834395422:1(243-262)Online publication date: 1-Jan-2000
https://dl.acm.org/doi/10.1137/S1064827598343954
Show More Cited By

A Stable Algorithm for Solving the Multifacility Location Problem Involving Euclidean Distances
1. Information systems
  1. Data management systems
    1. Database administration
2. Theory of computation
  1. Design and analysis of algorithms

Recommendations

On the Convergence of Miehle's Algorithm for the Euclidean Multifacility Location Problem

For the Euclidean single facility location problem, E. Weiszfeld proposed a simple iterative algorithm in 1937. Later, it was proved by numerous authors that it is a convergent descent algorithm. W. Miehle extended Weiszfeld's algorithm to solve the ...
The Euclidean Multifacility Location Problem

To solve the Euclidean multifacility location problem, W. Miehle extended the Weiszfeld algorithm from one to several facilities. L. M. Ostresh established that the objective function for the iterates of Miehle's algorithm decreases in value, but his ...
Minimax Multifacility Location with Euclidean Distances

The problem considered is that of locating N new facilities among M existing facilities with the objective of minimizing the maximum weighed Euclidean distance among all facilities. The application of nonlinear duality theory shows this problem can ...

Comments

Information & Contributors

Information

Published In

cover image SIAM Journal on Scientific and Statistical Computing

SIAM Journal on Scientific and Statistical Computing Volume 1, Issue 4

1980

126 pages

ISSN:0196-5204

Issue’s Table of Contents

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 December 1980

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

13
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 08 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

He SJiang YPan S(2010)An entropy regularization technique for minimizing a sum of Tchebycheff normsApplied Numerical Mathematics10.1016/j.apnum.2009.11.00560:3(208-223)Online publication date: 1-Mar-2010
https://dl.acm.org/doi/10.1016/j.apnum.2009.11.005
Pan SJiang Y(2008)Smoothing Newton Method for Minimizing the Sum of p-NormsJournal of Optimization Theory and Applications10.1007/s10957-008-9364-8137:2(255-275)Online publication date: 1-May-2008
https://dl.acm.org/doi/10.1007/s10957-008-9364-8
Andersen KChristiansen EConn AOverton M(2000)An Efficient Primal-Dual Interior-Point Method for Minimizing a Sum of Euclidean NormsSIAM Journal on Scientific Computing10.1137/S106482759834395422:1(243-262)Online publication date: 1-Jan-2000
https://dl.acm.org/doi/10.1137/S1064827598343954
Chan TGolub GMulet P(1999)A Nonlinear Primal-Dual Method for Total Variation-Based Image RestorationSIAM Journal on Scientific Computing10.1137/S106482759629976720:6(1964-1977)Online publication date: 1-Jan-1999
https://dl.acm.org/doi/10.1137/S1064827596299767
Conn AMongeau M(1998)Discontinuous piecewise linear optimizationMathematical Programming: Series A and B10.5555/3114207.311475480:3(315-380)Online publication date: 1-Feb-1998
https://dl.acm.org/doi/10.5555/3114207.3114754
Andersen KChristiansen EOverton M(1998)Computing Limit Loads by Minimizing a Sum of NormsSIAM Journal on Scientific Computing10.1137/S106482759427530319:3(1046-1062)Online publication date: 1-May-1998
https://dl.acm.org/doi/10.1137/S1064827594275303
Li Y(1998)A Newton Acceleration of the Weiszfeld Algorithm forMinimizing the Sum of Euclidean DistancesComputational Optimization and Applications10.1023/A:101833342241410:3(219-242)Online publication date: 1-Jul-1998
https://dl.acm.org/doi/10.1023/A%3A1018333422414
Burton DToint P(1992)On an instance of the inverse shortest paths problemMathematical Programming: Series A and B10.5555/3113610.311393253:1-3(45-61)Online publication date: 1-Jan-1992
https://dl.acm.org/doi/10.5555/3113610.3113932
Gurwitz C(1990)Weighted median algorithms for L1 approximationBIT10.1007/BF0201735030:2(301-310)Online publication date: 1-Jun-1990
https://dl.acm.org/doi/10.1007/BF02017350
Radó F(1988)The Euclidean Multifacility Location ProblemOperations Research10.5555/2804709.280471936:3(485-492)Online publication date: 1-Jun-1988
https://dl.acm.org/doi/10.5555/2804709.2804719
Show More Cited By

View Options

View options

Figures

Tables

Media

View Issue’s Table of Contents