Consistency of linear inequalities over sets

Author:
Abraham Berman

Journal:
Proc. Amer. Math. Soc. **36** (1972), 13-17

MSC:
Primary 15A39; Secondary 90C05

MathSciNet review:
0309967

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Abstract | References | Similar Articles | Additional Information

Abstract: Necessary and sufficient conditions for the equivalence of the statements: (I) The system , is consistent. (II) , are given in terms of the sets *S* and *T* and the matrix *A*. Sufficient conditions for this equivalence are obtained in the case where *S* and *T* are closed convex cones.

**[1]**Adi Ben-Israel,*Linear equations and inequalities on finite dimensional, real or complex, vector spaces: A unified theory*, J. Math. Anal. Appl.**27**(1969), 367–389. MR**0242865****[2]**Abraham Berman and Adi Ben-Israel,*More on linear inequalities with applications to matrix theory*, J. Math. Anal. Appl.**33**(1971), 482–496. MR**0279117****[3]**Abraham Berman and Adi Ben-Israel,*Linear inequalities, mathematical programming and matrix theory*, Math. Programming**1**(1971), 291–300. MR**0381716****[4]**Ky Fan,*A generalization of the Alaoglu-Bourbaki theorem and its applications*, Math. Z.**88**(1965), 48–60. MR**0178326****[5]**J. Farkas,*Über Theorie der einfachen Ungleichungen*, J. Reine Angew. Math.**124**(1902), 1-24.**[6]**Norman Levinson,*Linear programming in complex space*, J. Math. Anal. Appl.**14**(1966), 44–62. MR**0225569****[7]**R. Tyrrell Rockafellar,*Convex analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR**0274683****[8]**V. A. Sposito and H. T. David,*A note on Farkas lemmas over cone domains*, Iowa State University Report.

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0309967-6

Keywords:
Affine set,
closed convex cone,
consistency,
linear inequality,
linear transformation,
pointed cone,
polar,
polyhedral cone,
relative interior

Article copyright:
© Copyright 1972
American Mathematical Society