Abstract
Hačijan was the first one who succeeded in proving that a linear programming problem (or equivalently a system of linear inequalities) can be solved in polynomial time. Since it is well-known that this cannot be done by means of the simplex algorithm for linear programming, especially in the newspapers hopes were raised that the method used by Hačijan would result in a considerable improvement of the situation in numerical linear optimization. The aim of this paper is to discuss some numerical consequences of the new method and to relate it to other known approaches.
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References
V.Z. BELEN’KII and B.A. VOLKONSKIǐ, eds.: Iterative Methods in the Theory of Games and Programming. (Russian). Moscow: Izd. “Nauka” 1974.
K.-H. BORGWARDT: Some distribution-independent results about the asymptotic order of the average number of pivot steps in the simplex method. Universität Kaiserslautern, Fachbereich Mathematik, Preprint No. 4, March 1979.
L. COLLATZ and W. WETTERLING: Optimization Problems. Applied Mathematical Sciences, Vol. 17. Berlin, Heidelberg, New York: Springer-Verlag 1975.
D. DEGGIM: Auflösung linearer üngleichungssysteme durch Verfahren, die nicht auf Elimination beruhen. Diplomarbeit RWTH Aachen. 1972.
U. ECKHARDT: Iterative Lösung linearer Ungleichungssysteme. KPA Jülich Research Report Jül-880-MA, August 1972.
U. ECKHARDT: Ein Iterationsverfahren für lineare Ungleichungssysteme. Computing 12, 57–66 (1974).
U. ECKHARDT: Ein Iterationsverfahren zur Berechnung nicht-negativer Lösungen eines linearen Gleichungssystems. Z. Angew. Math.Mech. 54, T 213 — T 215 (1974).
P. GáCS and L. LOVASZ: Khachian’s algorithm for linear programming. Paper presented at the 10th International Symposium on Mathematical Programming, Montreal, August 27 – 31, 1979.
L.G. HAčIJAN: Polynomial algorithm in linear programming. (Russian). Dokl. Akad.Nauk SSSR 244, 1093 – 1096 (1979).
V. KLEE and G.J. MINTY: How good is the simplex algorithm? In: O. Shisha, ed.: Inequalities III, pp. 159 – 175. New York, London: Academic Press 1972.
O.L. MANGASARIAN: Nonlinear Programming. New York, St. Louis, San Francisco, London, Sydney, Toronto, Mexico, Panama: McGraw-Hill Book Company 1969.
O.L. MANGASARIAN: Iterative solution of linear programs. University of Wisconsin — Madison, Computer Sciences Technical Report Nr. 327, June 1979.
E.H. McCALL: A study of the Khachian algorithm for real-world linear programming. Sperry Univac, P.O.Box 43942, St. Paul, Minnesota 55164, January 1980.
N.Z. ŠOR: On the speed of convergence of the method of generalized gradient descent with stretching of the space. (Russian). Kibernetika (Kiev) 1970, No.2, 80–85.
N.Z. ŠOR and N.G. ŽURBENKO: A minimization method using the operation of stretching of the space in direction of the difference of two successive gradients. (Russian). Kibernetika (Kiev) 1971, No.3, 51 – 59.
R. WESTPHAL: Iterative Lösung linearer Ungleichungssysteme. Diplomarbeit Universität Hamburg, 1980.
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Eckhardt, U. (1980). Some Remarks on Hačijan’s Paper. In: Collatz, L., Meinardus, G., Wetterling, W. (eds) Konstruktive Methoden der finiten nichtlinearen Optimierung. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 55. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6322-3_4
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DOI: https://doi.org/10.1007/978-3-0348-6322-3_4
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