Abstract
Versions of linear programming in abstract spaces have been formulated by Duffin [5], Hurewicz [8], Kretschmer [IO], Fan [6], and Ben-Israel, Charnes, and Kortanek (BCK) [2, 31. In this paper, yet another, rather geometric, formulation of linear programming is presented in which the duality theorem of classical linear programming holds almost verbatim (Theorem 1). A sufficient condition for the existence of optimal solutions is also given (Theorem 2). The duality and existence theorems of Duffin’s formulation of linear programming [5] are shown to follow from these two theorems. The formulation of linear programming presented here may be viewed as a special case of that of BCK [2]; in fact, Kortanek and Soyster have shown [7] that the classification scheme of [2] gives a refinement of Theorem 1 of this paper. Going the other way, Theorem 1 of this paper provides a short proof of the Ben-Isreal and Charnes solvability theorem [l] which is an essential tool in proving the classification theorem of [2]. Finally, the formulation of linear programming presented here has been used in proving a duality and existence theorem in continuous linear programming [ 1 I].
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