From the Farkas Lemma to the Hahn--Banach Theorem

Abstract

This paper provides new versions of the Farkas lemma characterizing those inequalities of the form $f(x)\geq 0$ which are consequences of a composite convex inequality $(S\circ g)(x)\leq 0$ on a closed convex subset of a given locally convex topological vector space $X$, where $f$ is a proper lower semicontinuous convex function defined on $X,$ $S$ is an extended sublinear function, and $g$ is a vector-valued $S$-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of $f$, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as equivalent to an extended version of the so-called Hahn--Banach--Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn--Banach t...