On semi-infinite systems of linear inequalities

The ordered fieldR(M) consists of the realsR with a transcendentalM adjoined, which is larger than any realr ∈R. Given any semi-infinite matrix (s.i.m.) interpreted as linear inequalities:u ^tP_i≧c _i, ∀ _i ∈I, an arbitrary index set, it is also shown that the following are equivalent. (1) For every finiteJ ⊆I the systemu ^tP_i≧c _i,i ∈J is consistent, and (2) the s.i.m. has a solutionu ∈R(M) ⁿ. Some consequences for “duality gaps” are also given. These results were obtained as part of the activities of the Management Science Research Group and School of Urban and Public Affairs, Carnegie-Mellon University.