Farkas' Lemma, other theorems of the alternative, and linear programming in infinite-dimensional spaces: a purely linear-algebraic approach

References

• Farkas , J . 1901 . Theorie der einfachen Ungleichungen. . Journal für die Reine and Angewandte Mathamatik , 124 : 1 – 27 .
   Google Scholar
• Lukeš , J . 1998 . Zápisky z funkcionální analýzy (Manuscripts on Functional Analysis, in Czech) , Praha : Karolinum . ISBN 80-7184-597-3
   Google Scholar
• Fan , K . 1956 . “ On systems of linear inequalities ” . In Linear Inequalities and Related Systems , Edited by: Kuhn , HW and Tucker , AW . 99 – 156 . Princeton : Princeton University Press .
   Google Scholar
• (in Russian); Chernikov, S.N., 1968, Linear Inequalities (Moskva: Nauka).
   Google Scholar
• Broyden , CG . 1998 . A simple algebraic proof of Farkas's Lemma and related theorems . Optimization Methods and Software , 8 : 185 – 199 .
   Web of Science ®Google Scholar
• Tucker , AW . 1956 . “ Dual systems of homogeneous linear relations ” . In Linear Inequalities and Related Systems , Edited by: Kuhn , HW and Tucker , AW . 3 – 18 . Princeton : Princeton University Press .
   Google Scholar
• Good , RA . 1959 . Systems of linear relations. . SIAM Review , 1 : 1 – 31 .
   Web of Science ®Google Scholar
• Dax , A . 1993 . The relationship between theorems of the alternative, least norm problems, steepest descent directions, and degeneracy: a review. . Annals of Operations Research , 46 : 11 – 60 .
   Google Scholar
• Dax , A and Sreedharan , VP . 1997 . Theorems of the alternative and duality. . Journal of Optimization Theory and Applications , 94 : 561 – 590 .
   Web of Science ®Google Scholar
• Dax , A . 1997 . An elementary proof of Farkas' Lemma. . SIAM Review , 39 : 503 – 507 .
   Web of Science ®Google Scholar
• Roos , C and Terlaky , T . 1999 . Note on a paper of Broyden. . Operations Research Letters , 25 : 183 – 186 .
   Google Scholar
• Prékopa , A . 1980 . On the development of optimization theory. . The American Mathematical Monthly , 87 : 527 – 542 .
   Web of Science ®Google Scholar
• Greenberg , HJ . c1996–2006 . Mathematical programming glossary supplement: alternative systems , Available online at: http://glossary.computing.society.informs.org/second.php?page=alterntv.html (accessed June 2006)
   Google Scholar
• Fan , K , Glicksberg , I and Hoffman , AJ . 1957 . Systems of inequalities involving convex functions. . Proceedings of American Mathematical Society , 8 : 617 – 622 .
   Google Scholar
• Franklin , J . c1980 . Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems , New York : Springer . ISBN 0-387-90481-6
   Google Scholar
• Dantzig , GB , Orden , A and Wolfe , P . 1955 . The generalized simplex method for minimizing a linear form under linear inequality restraints. . Pacific Journal of Mathematics , 5 : 183 – 195 .
   Google Scholar
• Fredholm , I . 1903 . Sur une classe d'équations fonctionnelles. . Acta Mathematica , 27 : 365 – 390 .
   Google Scholar
• Gale , D . 1960 . The Theory of Linear Economic Models , New York : McGraw-Hill .
   Google Scholar
• Motzkin , TS . 1934 . Beiträge zur Theorie der linearen Ungleichungen , Jerusalem : Azriel . Doctoral dissertation, University of Basel.,1936
   Google Scholar
• Motzkin , TS . 1952 . “ Contributions to the theory of linear inequalities, D.R. Fulkerson (Transl.) (Santa Monica: RAND Corporation). RAND Corporation Translation 22 ” . In Theodore S. Motzkin: Selected Papers , Edited by: Cantor , D , Gordon , B and Rothschild , B . 1 – 80 . Boston : Birkhäuser, 1983 . ISBN 3-7643-3087-2
   Google Scholar
• Gordan , P . 1873 . Ueber die Auflösung linearer Gleichungen mit reellen Coefficienten. . Mathematische Annalen , 6 : 23 – 28 .
   Google Scholar
• Stiemke , E . 1915 . Über positive Lösungen homogener linearer Gleichungen. . Mathematische Annalen , 76 : 340 – 342 .
   Google Scholar
• Carver , WB . 1923 . Systems of linear inequalities. . Annals of Mathematics , 23 : 212 – 220 .
   Google Scholar