Alternative systems
From Glossary
Two systems of inequalities and/or equations such that there exists a solution
to one system, or there exists a solution to the other system, and there cannot
exist a solution to both systems. Several alternative systems are widely
used in mathematical programming, especially in the study of duality.
 Fourier (1826)
 (not vacuous)
 or
 (not vacuous)
 Gordan (1873)

 or

 Farkas (1902)

 or
 A Variant is

 or

 Fredholm (1903)

 or

 Stiemke (1915)

 or

 Motzkin (1936)
 ( not vacuous)
 or
 ( not vacuous)
 Ville (1938)

 or

 Tucker (1956)

 or

 Gale (1960)

 or

[edit]
Integer Systems
 Papadimitriou and Steiglitz (1982)
Let be an integer matrix, ,
, and an integer vector.

 or
 .

[edit]
Convex Systems
For the following, is convex on .
 Fan, Glicksburg and Hoffman (1957)

 or
 for all .
