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A note on optimality conditions for the Euclidean multifacility location problem.(English)Zbl 0614.90033

The key problem of the Euclidean multifacility location (EMFL) problem is to decide whether a given dead point is optimal. If it is not optimal, we wish to compute a descent direction. This paper extends the optimality conditions ofP. H. Calamai andA. R. Conn [SIAM J. Sci. Stat. Comput. 1, 512-526 (1980;Zbl 0469.65040); Lect. Notes Math. 912, 1-25 (1982;Zbl 0485.65013)] andM. L. Overton [Math. Program. 27, 34-63 (1983;Zbl 0536.65053)] to the case when the rows of the active constraints matrix are linearly dependent. We show that linear dependence occurs whenever G, the graph of the coinciding facilities, has a cycle. In this case the key problem is formulated as a linear least squares problem with bounds on the Euclidean norms of certain subvectors.

MSC:

90B05 Inventory, storage, reservoirs

Cite

References:

[1]R.G. Busacker and T.L. Saaty,Finite Graphs and Networks (McGraw-Hill, New York, 1965). ·Zbl 0146.20104
[2]S. Busovaca, ”Handling degeneracy in a nonlinearl 1 algorithm,” Technical Report CS85 34, University of Waterloo (Waterloo, Ontario, 1985).
[3]P.H. Calamai, ”On numerical methods for continuous location problems,” Ph.D. Thesis, University of Waterloo (Waterloo, Ontario, 1983).
[4]P.H. Calamai and C. Charalambous, ”Solving multifacility location problems, involving Euclidean distances,” Naval Research Logistics Quarterly 27 (1980) 609–620. ·Zbl 0447.90022 ·doi:10.1002/nav.3800270408
[5]P.H. Calamai and A.R. Conn, ”A stable algorithm for solving the multifacility location problem involving Euclidean distances,” SIAM Journal on Scientific and Statistical Computing 1 (1980) 512–526. ·Zbl 0469.65040 ·doi:10.1137/0901037
[6]P.H. Calamai and A.R. Conn, ”A second-order method for solving the continuous multifacility location problem,” in: G.A. Watson, ed.,Numerical Analysis: Proceedings of the Ninth Biennial Conference, Dundee, Scotland, Lecture Notes in Mathematics 912 (Springer-Verlag, Berlin, 1982), pp. 1–25. ·Zbl 0485.65013
[7]A. Dax, ”An efficient algorithm for solving the rectilinear multifacility location problem” submitted for publication in IMA Journal of Numerical Analysis (1984).
[8]A. Dax, ”An efficient clustering technique for solving the Euclidean multifacility location problem,” in preparation (1986). ·Zbl 0632.65074
[9]M.L. Overton, ”A quadratically convergent method for minimizing a sum of Euclidean norms,” Mathematical Programming 27 (1983) 34–63. ·Zbl 0536.65053 ·doi:10.1007/BF02591963
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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