The Farkas Lemma Revisited

The Farkas Lemma is extended to simultaneous linear operator and polyhedral sublinear operator inequalities by Boolean valued analysis. Introduction The Farkas Lemma, also known as the Farkas–Minkowski Lemma, plays a key role in linear programming and the relevant areas of optimization (cp. [1]). Some rather simple proof of the lemma is given in [2]. Using the technique of Boolean valued analysis (cp. [3]), we abstract the Farkas Lemma to simultaneous linear and polyhedral sublinear inequalities with operators. Assume that X is a real vector space, Y is a Kantorovich space also known as a Dedekind complete vector lattice or a complete Riesz space. Let B := B(Y ) be the base of Y , i.e., the complete Boolean algebra of positive projections in Y ; and let m(Y ) be the universal completion of Y . Denote by L(X,Y ) the space of linear operators from X to Y . In case X is furnished with some Y -seminorm, by L(X,Y ) we mean the space of dominated operators from X to Y . As usual, {T ≤ 0} := {x ∈ X ∣ Tx ≤ 0} and {T = 0} := ker(T ) := T−1(0) for T : X → Y . 1. Simultaneous Linear Inequalities The Farkas Lemma deals with the following diagram: