Abstract
The conventional procedure for folding a system of linear inequalities based on the Fourier-Chernikov algorithm is supplemented with techniques for eliminating redundant inequalities, which considerably counteracts the increase in the system dimension. Exact and approximate methods are proposed, which are brought to algorithmic form and software implementation. Numerical results are discussed.
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References
D. V. Shapot, “On the Construction of Orthogonal Projections of Point Sets Defined by a System of Inequalities,” Zh. Vychisl. Mat. Mat. Fiz. 11, 1113–1126 (1971).
T. S. Motzkin, H. Raiffa, G. L. Tompson, and R. M. Thrall, “Double Description Method,” in Contributions to the Theory of Games. Annals of Mathematics Study, Ed. By H. W. Kuhn and A. W. Tucker (Princeton Univ. Press, Princeton, 1953; Fizmatgiz, Moscow, 1961), Vol 2, No. 28, pp. 51–73.
S. N. Chernikov, Linear Inequalities (Nauka, Moscow, 1968) [in Russian].
D. V. Shapot and A. M. Lukatskii, “A Constructive Algorithm for Building Orthogonal Projections of Polyhedral Sets,” in Optimization, Control, and Intelligence (SEI, Siberian Division, Russian Academy of Sciences, Irkutsk, 1995), No. 1, pp. 77–91 [in Russian].
V. A. Bushenkov and A. V. Lotov, “An Algorithm for the Analysis of Inequality Dependences in a Linear System,” Zh. Vychisl. Mat. Mat. Fiz. 20, 562–572 (1980).
A. V. Lotov, “On the Estimate of Stability and the Condition Number of the Solution Set of a System of Linear Inequalities,” Zh. Vychisl. Mat. Mat. Fiz. 24, 1763–1774 (1984).
I. I. Eremin and A. A. Makhnev, “International Workshop on Algebra and Linear Optimization Dedicated to the 90th Anniversary of the Birth of S. N. Chernikov,” Izv. Ural’skogo Gos. Univ., No. 30, 183–184 (2004).
R. V. Efremov, G. K. Kamenev, and A. V. Lotov, “Constructing an Economical Description of a Polytope Using the Duality Theory of Convex Sets,” Dokl. Akad. Nauk 399(5), 594–596 (2004) [Dokl. Math. 70, 934–936 (2004)].
A. I. Golikov and Yu. G. Evtushenko, “A New Method for Solving Systems of Linear Equalities and Inequalities,” Dokl. Akad. Nauk 381, 444–447 (2001) [Dokl. Math. 64, 370–373 (2001)].
P. O. Gutman and I. Ioslovich, “On the Generalized Wolfe Problem: Preliminary Analysis of Nonnegativity of Large Linear Programs Subject to Group Constraints,” Avtom. Telemekh., No. 8, 116–125 (2007).
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Original Russian Text © A.M. Lukatskii, D.V. Shapot, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 7, pp. 1167–1180.
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Lukatskii, A.M., Shapot, D.V. A constructive algorithm for folding large-scale systems of linear inequalities. Comput. Math. and Math. Phys. 48, 1100–1112 (2008). https://doi.org/10.1134/S0965542508070038
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DOI: https://doi.org/10.1134/S0965542508070038