Consistency of linear inequalities over sets
Author:
Abraham Berman
Journal:
Proc. Amer. Math. Soc. 36 (1972), 1317
MSC:
Primary 15A39; Secondary 90C05
MathSciNet review:
0309967
Fulltext PDF Free Access
Abstract 
References 
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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
BenIsrael, 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
(39 #4192)
 [2]
Abraham
Berman and Adi
BenIsrael, More on linear inequalities with applications to matrix
theory, J. Math. Anal. Appl. 33 (1971),
482–496. MR 0279117
(43 #4843)
 [3]
Abraham
Berman and Adi
BenIsrael, Linear inequalities, mathematical programming and
matrix theory, Math. Programming 1 (1971),
291–300. MR 0381716
(52 #2605)
 [4]
Ky
Fan, A generalization of the AlaogluBourbaki theorem and its
applications, Math. Z. 88 (1965), 48–60. MR 0178326
(31 #2584)
 [5]
J. Farkas, Über Theorie der einfachen Ungleichungen, J. Reine Angew. Math. 124 (1902), 124.
 [6]
Norman
Levinson, Linear programming in complex space, J. Math. Anal.
Appl. 14 (1966), 44–62. MR 0225569
(37 #1162)
 [7]
R.
Tyrrell Rockafellar, Convex analysis, Princeton Mathematical
Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
(43 #445)
 [8]
V. A. Sposito and H. T. David, A note on Farkas lemmas over cone domains, Iowa State University Report.
 [1]
 A. BenIsrael, Linear equations and inequalities on finite dimensional, real or complex, vector spaces: A unified theory, J. Math. Anal. Appl. 27 (1969), 367389. MR 39 #4192. MR 0242865 (39:4192)
 [2]
 A. Berman and A. BenIsrael, More on linear inequalities with applications to matrix theory, J. Math. Anal. Appl. 33 (1971), 482496. MR 0279117 (43:4843)
 [3]
 , Linear inequalities, mathematical programming and matrix theory, Math. Prog. 1 (1971), 291300. MR 0381716 (52:2605)
 [4]
 K. Fan, A generalization of the AlaogluBourbaki theorem and applications, Math. Z. 88 (1965), 4860. MR 31 #2584. MR 0178326 (31:2584)
 [5]
 J. Farkas, Über Theorie der einfachen Ungleichungen, J. Reine Angew. Math. 124 (1902), 124.
 [6]
 N. Levinson, Linear programming in complex space, J. Math. Anal. Appl. 14 (1966), 4462. MR 37 #1162. MR 0225569 (37:1162)
 [7]
 R. T. Rockafellar, Convex analysis, Princeton Math. Series, no. 28, Princeton Univ. Press, Princeton, N.J., 1970. MR 43 #445. MR 0274683 (43:445)
 [8]
 V. A. Sposito and H. T. David, A note on Farkas lemmas over cone domains, Iowa State University Report.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197203099676
PII:
S 00029939(1972)03099676
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
