Abstract
This paper describes a method and the corresponding algorithms for simplification of large-scale linear programming models. It consists of the elimination of the balance constraints (i.e. constraints with zero RHS term). The idea is to apply some linear transformations to the original problem in order to nullify the balance constraints. These transformations are able to simultaneously eliminate more balance rows. The core of this contribution is the introduction of the reduction matrix and the associated theorems on the equivalent linear programs (original and reduced). The numerical experiments with this method of simplification proved this approach to be beneficial for a large class of LP problems.
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References
N. Andrei, C. Rasturnoiu and M. Barbulescu, Towards an interactive and integrated system for large-scale optimization, Technical Report LSSO-89-1, Research Institute for Informatics, Analysis and Mathematical Modeling of Systems Laboratory, Bucharest (1989).
N. Andrei, C. Rasturnoiu and M. Barbulescu, ISLO — Interactive System for Linear Optimization, Technical Report LSSO-90-1, Research Institute for Informatics, Analysis and Mathematical Modeling of Systems Laboratory, Bucharest (1990).
N. Andrei and C. Rasturnoiu,Sparse Matrices and their Applications (Technical Press, Bucharest, 1983).
A.L. Brearley, G. Mitra and H.P. Williams, Analysis of mathematical programming problems prior to applying the simplex algorithm, Math. Progr. 8(1975)54–83.
H.J. Greenberg, A functional description of ANALYZE: A computer-assisted analysis system for linear programming models, ACM TOMS 9(1983)18–56.
H.J. Greenberg and J.S. Maybee (eds.),Proc. 1st Symp. on Computer-Assisted Analysis and Model Simplification, University of Colorado, Boulder, CO (Academic Press, 1981).
H.J. Greenberg, F. Murphy and S.H. Shaw (eds.),Advanced Techniques in the Practice of Operations Research (Elsevier, New York, 1982).
H.J. Greenberg, ANALYZE rulebase, in:Mathematical Models for Decision Support, ed. G. Mitra, NATO-ASI Series (Springer, Berlin, 1988).
G.F. Knolmayer, Balance equation and modeling principles,6th Annual Int. Mathematical Programming Symp. Brussels (1976).
G.F. Knolmayer, Computational experiments in the formulation of large-scale linear programs, in:Large-Scale Linear Programming, ed. G. Dantzig, M.H. Dempster and M. Kallio, Proc. IIASA Workshop (1980) vol. 2, pp. 865–887.
G.F. Knolmayer,Programmierungsmodelle für die Produktions Programmplanung — Ein Beitrag zur Methodologie der Modelkonstruktion (Birkhäuser, 1980).
G.F. Knolmayer, Computational experiments in the formulation of linear product-mix and non-convex production-investment models, Comp. Oper. Res. 9(1982)207–219.
H.P. Williams,Model Building in Mathematical Programming, 2nd ed. (Wiley-Interscience, Chichester, 1985).
U. Zimmermann, On recent developments in linear programming, in:Trends in Mathematical Optimization, 4th French-German Conf. on Optimization, ed. K.H. Hoffmann, J.B. Hiriart-Urruty, C. Lemarechal and J. Zowe, Irsee (Birkhäuser, Basel-Boston, 1988) pp. 353–390.
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This research work was done while the first author was at Duisburg University, Mess-, Steuer und Regelungstechnik, Germany, under the greatly appreciated financial assistance given by the Alexander-von-Humboldt Foundation.
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Andrei, N., Barbulescu, M. Balance constraints reduction of large-scale linear programming problems. Ann Oper Res 43, 147–170 (1993). https://doi.org/10.1007/BF02025015
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DOI: https://doi.org/10.1007/BF02025015