A simple algebraic proof of Farkas's lemma and related theorems

A proof is given of Farkas's lemma based on a new theorem pertaining to orthogodal matrices. It is claimed that this theorem is slightly more general than Tucker's theorem, which concerns skew-symmetric matrices and which may itself be derived simply from tne new theorem. Farkas's lemma and other theorems of the alternative then follow trivially from Tucker's theorem

[1]  Steven Vajda,et al.  The Theory of Linear Economic Models , 1960 .

[2]  Robert G. Bland,et al.  A combinatorial abstraction of linear programming , 1977, J. Comb. Theory B.

[3]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[4]  P. V. Remoortere.  Linear and combinatorial programming : K.G. Murty: 1976,J. Wiley, New York, 310 pp , 1979 .

[5]  Hanif D. Sherali,et al.  Linear Programming and Network Flows , 1977 .

[6]  J. Farkas.  Theorie der einfachen Ungleichungen. , 1902 .

[7]  V. P. Sreedharan,et al.  Theorems of the Alternative and Duality , 1997 .

[8]  Skew-symmetric matrices, staircase functions and theorems of the alternative , 1986 .

[9]  A. Tucker.  1 . Dual Systems of Homogeneous Linear Relations , 1957 .

[10]  G. Ziegler.  Lectures on Polytopes , 1994 .

[11]  Antal E. Fekete,et al.  Real Linear Algebra , 1985 .

[12]  L. Dixon,et al.  Finite Algorithms in Optimization and Data Analysis. , 1988 .

[13]  Richard A. Tapia,et al.  Practical Methods of Optimization, Volume 2: Constrained Optimization (R. Fletcher) , 1984 .

[14]  Achiya Dax,et al.  The relationship between theorems of the alternative, least norm problems, steepest descent directions, and degeneracy: A review , 1993, Ann. Oper. Res..

[15]  A. Dax,et al.  AN ELEMENTARY PROOF OF FARKAS ’ LEMMA , 1997 .

[16]  C. G. Broyden,et al.  Some LP algorithms using orthogonal matrices , 1995 .

[17]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[18]  Steven Vajda,et al.  Mathematical programming , 1961, Addison-Wesley series in statistics.

[19]  R. A. Good.  Systems of Linear Relations , 1959 .

[20]  András Prékopa,et al.  On the Development of Optimization Theory , 1980 .

[21]  Achiya Dax,et al.  Classroom Note: An Elementary Proof of Farkas' Lemma , 1997, SIAM Rev..