Generalizations of Farkas’ Theorem

Abstract

A unified treatment is given of generalizations of Farkas' theorem on linear inequalities to arbitrary convex cones and to dual pairs of real vector spaces of arbitrary dimension. Various theorems for locally convex spaces readily follow. The results are applied to duality and converse duality theory for linear programming and to a generalization of the Kuhn-Tucker theorem, both of these in spaces of arbitrary dimension and with inequalities involving arbitrary convex cones.1. Introduction. Farkas' theorem on systems of linear inequalities 11] (see also [ 10, Thm. 4-1) has been generalized to systems involving polyhedral cones [ 1], and to arbitrary cones in locally convex spaces [12]. A unified theory is presented here, based on arbitrary cones and dual pairs of real vector spaces. A necessary and sufficient condition is obtained (Theorem 2) for the solvability of a linear equation Ax b by a vector x in a given convex cone. A related necessary and sufficient condition (Theorem 1) is obtained as an inclusion relation between two cones. Although these results are closely related to the Hahn-Banach separation

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