Abstract
Fourier treated a system of linear inequalities by a method of elimination of variables. This method can be used to derive the duality theory of linear programming. Perhaps this furnishes the quickest proof both for finite and infinite linear programs. For numerical evaluation of a linear program, Fourier’s procedure is very cumbersome because a variable is eliminated by adding each pair of inequalities having coefficients of opposite sign. This introduces many redundant inequalities. However, modifications are possible which reduce the number of redundant inequalities generated. With these modifications the method of Fourier becomes a practical computational algorithm for a class of parametric linear programs.
Prepared under Research Grant DA-AROD-31-124-71-G17, Army Research Office, Durham.
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References
J.B.J. Fourier, “Solution d’une question particulière du calcul des inégalités”, in: Oeuvres II (Paris, 1890).
L.L. Dines, “Concerning preferential voting”, American Mathematical Monthly 24 (1917) 321–325.
L.L. Dines, “Systems of linear inequalities”, Annals of Mathematics 20, (1919) 191–199.
L.L. Dines, “Definite linear dependence”, Annals of Mathematics 27 (1925) 57–64.
L.L. Dines, “On positive solutions of a system of linear equations”, Annals of Mathematics 28 (1927) 386–392.
L.L. Dines and N.H. McCoy, “On linear inequalities”, Transactions of the Royal Society of Canada 27 (1933) 37–70.
T.S. Motzkin, “Beiträge zur Theorie der linearen Ungleichungen”, Dissertation, University of Basel (Jerusalem, 1936).
H.W. Kuhn, “Solvability and consistency for linear equations and inequalities”, American Mathematical Monthly (1956) 217–232.
S.N. Chernikov, “The solution of linear programming problems by elimination of unknowns”, Doklady Akademii Nauk SSSR 139 (1961) 1314–1317. [Translation in: Soviet Mathematics Doklady 2 (1961) 1099–1103.]
S.N. Chernikov, “Contraction of finite systems of linear inequalities”, Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 5 (1965) 3–20.
J. Abadie, “The dual to Fourier’s method for solving linear inequalities”, International Symposium on Mathematical Programming, London, 1964.
R.J. Duffin, “An orthogonality theorem of Dines related to moment problems and linear programming”, Journal of Combinatorial Theory 2 (1967) 1–26.
D.A. Kohler, “Projections of convex polyhedral sets”, Operations Research Center Rept. 67-29, University of California, Berkeley, Calif. (1967).
D.A. Kohler, “Translation of a report by Fourier on his work on linear inequalities”, Opsearch 10 (1973) 38–42.
G.B. Dantzig and B.C. Eaves, “Fourier-Motzkin elimination and its dual”, Dept. of Operations Research Rept., Stanford University, Stanford, Calif. (January 1973).
V. Klee, “Vertices of convex polytopes”, Lecture Notes, Department of Mathematics, University of Washington, Seattle, Wash. (1973).
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Dedicated to Albert Tucker who has made such instructive use of tableaus simultaneously depicting primal and dual programs. This is the inspiration for the tableau of the present paper which simultaneously depicts Fourier elimination and its dual, Dines elimination.
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© 1974 The Mathematical Programming Society
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Duffin, R.J. (1974). On fourier’s analysis of linear inequality systems. In: Balinski, M.L. (eds) Pivoting and Extension. Mathematical Programming Studies, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121242
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DOI: https://doi.org/10.1007/BFb0121242
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