Abstract
Recently, linear programming problems with special structures have assumed growing importance in mathematical programming. It is well known that exploiting network structures within linear programs can lead to considerable improvement of the computational solution of large-scale linear programming problems. A linear program is said to contain an embedded network structure provided that some subset of its constraints can be interpreted as specifying conservation of flow. If a column of the constraint matrix has at most two non-zeros, then it leads to embedded generalized network structure and if these non-zeros are unit elements and of opposite signs, then it leads to embedded pure network structure.
In this paper, we are concerned with algorithms for detecting embedded pure network structures within linear programs. The network extraction methods are presented in two groups. The first group covers deletion and addition based algorithms and the second group covers GUB based algorithms. We have extended the GUB based algorithm appearing in the second group by introducing Markowitz merit count approach for exploiting matrix non zeros. A set of well known test problems has been used to carry out computational experiments which show that our extensions to the GUB based algorithms give better results than the algorithms reported earlier.
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References
Baker, B. M., Maye, P.J. (1993). A Heuristic for Finding Embedded Network Structure in Mathematical Programs,European Journal of Operational Research,67, pp. 52–63.
Baston, V.J.D., Rahmouni, M.K., Williams, H.P. (1991), The Practical Conversion of Linear Programmes to Network Flow Models,European Journal of Operational Research,50, pp. 325–334.
Birge, J.R., Dempster, M.A.H., Gassman, H.I., Gunn, E.A., King, A.J., Wallace, S. (1987), A Standart Input Format for Multiperiod Stochastic Linear Programs,Mathematical Programming Society, Coal, Newsletter, Vol.17.
Bixby, R.E., Fourer, R. (1988), Finding Embedded Network Rows in Linear Programs I. Extraction Heuristics,Management Science, Vol.34, No. 3, pp. 342–376.
Bixby, R.E., Fourer, R. (1995), Finding Embedded Network Rows in Linear Programs II. Augmentation Heuristics, unpublished.
Bixby, R.E. (1981), Hidden Structures in Linear Programs, inComputer Assisted Analysis and Model Simplification, (Ed.) Greenberg, H., Maybee, J., Academic Press, New York, pp. 327–360.
Bixby, R.E., Cunningham, H.W. (1980), Converting Linear Programs to Network Problems,Mathematics of Operations Research, Vol.5, No. 3, pp. 321–356.
Brearley, A.L., Mitra, G., Williams, H.P. (1975), Analysis of Mathematical Programming Problems Prior to Applying the Simplex Algorithm,Mathematical Programming,8, pp. 54–83.
Brown, G.G., Wright, W.G. (1984), Automatic Identification of Embedded Network Rows in Large-Scale Optimization Models,Mathematical Programming,29, pp. 41–56.
Brown, G.G., McBride, R.D., Wood, R.K. (1985), Extracting Embedded Generalized Networks From Linear Programming Problems,Mathematical Programming,32, pp. 11–31.
Brown, G.G., Wright, W.G. (1981), Automatic Identification of Embedded Structure in Large-Scale Optimization Models, inComputer Assisted Analysis and Model Simplification, (Ed.) Greenberg, H., Maybee, J., Academic Press, New York, pp. 369–388.
Brown, G.G., Thomen, D.S. (1980), Automatic Identification of Generalized Upper Bounds in Large Scale Optimization Models,Management Science, Vol.26, No. 11, pp. 1166–1184.
Dantzig, G.B., Van Slyke, R.M. (1967), Generalized Upper Bounding Techniques,Journal of Computer and System Science, Vol.1, pp. 213–226.
Dobson, G. (1982), Worst Case Analysis of Greedy Heuristics For Integer Programming with Nonnegative Data,Mathematics of Operations Research, Vol.7, No. 4, pp. 515–531.
Duff, S., Erisman, A.M., Reid, J.K. (1986),Direct Methods for Sparse Matrices, Oxford University Press, Oxford, London.
Gay, D.M. (1985), Electronic Mail Distribution of Linear Programming Test Problems,Mathematical Programming Society Coal, Newsletter, Vol.13, pp. 10–12.
Glover, F. (1981), Creating Network Structures in Linear Programs, inComputer Assisted Analysis and Model Simplification, (Ed.) Greenberg, H., Maybee, J., Academic Press, New York, pp. 361–367.
Glover, F., Klingman, D. (1971), The Simplex SON Algorithm For LP/Embedded Network Problems,Mathematical Programming Study,15, pp. 148–176.
Greenberg, J. H., Rarick, C. D. (1974), Determining GUB Sets Via an Invert Agenda Algorithm,Mathematical Programming,7, pp. 240–244.
Gülpinar, N., Mitra, G., Maros, I. (1996), Detecting Embedded Network Structures in Large Scale LP Problems, Technical Report, TR/20/1996, Brunel University.
Gunawardane, G., Hoff, S., Schrage, L. (1981), Identification of Special Structure Constraints in Linear Programs,Mathematical Programming,21, pp. 90–97.
Kennington, J., Helgason, R. (1980),Algorithms for Network Programming, John Wiley and Sons, New York, NY.
Klingman, D. (1977), Finding Equivalent Network Formulations For Constrained Network Problems,Management Science, Vol.23, No. 7, pp. 737–744.
Markowitz, H.M. (1957), The Elimination Form of the Inverse and Its Application to Linear Programming,Management Science, Vol.3, pp 255–267.
Messina, E., Mitra, G. (1996), Modelling and Analysis of Multi-stage Stochastic Programming Problems: A Software Environment, Technical Report, TR/03/1996, Brunel University.
Mitra, G., Tamiz, M. (1991), Alternative Methods For Representing the Inverse of Linear Programming Basis Matrices, inRecent Developments in Mathematical Programming, (Ed.) Kumar, S., pp. 273–301.
Schrage, L. (1981), Some Comments on Hidden Structure in Linear Programs, inComputer Assisted Analysis and Model Simplification, (Ed.) Greenberg, H., Maybee, J., Academic Press, New York, pp. 389–395.
Schrage, L. (1978), Implicit Representation of Generalized Variable Upper Bounds in Linear Programming,Mathematical Programming,14, pp. 11–20.
Toyoda, Y. (1975), Simplified Algorithm For Obtaining Approximate Solutions to Zero-One Programming Problems,Management Science, Vol.21, No. 12, pp. 1417–1427.
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Gülpinar, N., Mitra, G. & Maros, I. Detecting embedded pure network structures in LP problems. Top 6, 67–95 (1998). https://doi.org/10.1007/BF02564799
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DOI: https://doi.org/10.1007/BF02564799