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The Relaxation Method for Linear Inequalities

Published online by Cambridge University Press:  20 November 2018

Shmuel Agmon*
Affiliation:
The Rice Institute, Houston, Texas, University of California at Los Angeles; National Bureau of Standards at Los Angeles, The Hebrew University, Jerusalem
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In various numerical problems one is confronted with the task of solving a system of linear inequalities:

(1.1) (i = 1, … ,m)

assuming, of course, that the above system is consistent. Sometimes one has, in addition, to minimize a given linear form l(x). Thus, in linear programming one obtains a problem of the latter type.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Dantzig, G. B., Maximization of a linear form whose variables are subject to a system of linear inequalities (U.S.A.F., 1949), 16 pp.Google Scholar
2. Forsythe, G. E., Solving linear algebraic equations can be interesting, Bull. Amer. Math. Soc, 59 (1953), 299329.CrossRefGoogle Scholar
3. Motzkin, T. S. and Raiffa, H., Thompson, G. L., Thrall, R. M., The double description method, in Contributions to the Theory of Games, Annals of Mathematics Series, 2 (1953), 5174.Google Scholar
4. Southwell, R. V., Relaxation methods in engineering science (Oxford, 1940).Google Scholar
5. Southwell, R. V., Relaxation methods in theoretical physics (Oxford, 1946).Google Scholar
6. Temple, G., The general theory of relaxations applied to linear systems, Proc. Roy. Soc. London, 169 (1939), 476500.Google Scholar
7. Linear programming seminar notes, Institute for Numerical Analysis (Los Angeles, 1950).Google Scholar
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