From the Farkas Lemma to the Hahn--Banach Theorem

Abstract

Abstract. This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f (x) ≥ 0 which are consequences of a composite convex inequality (S • g)(x) ≤ 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 theorem and the Mazur-Orlicz theorem for extended sublinear functions.

Farkas-Type Results for Vector-Valued Functions with Applications

“…In the rest of this section we consider some special cases where the results above collapse to several well-known asymptotic Farkas-type results in the literature ( [6], [9]). These results have been used to get optimality conditions, duality theorems, and set containment characterizations for (SOP).…”

“…The next stable Farkas lemma for convex systems under linear perturbations [6] is a direct consequence of the previous results. [6] Let (f; g; C) be a scalar triple such that f 2 (X), the convex cone S is closed, and g is S-convex and S-epi closed.…”

“…[6] Let (f; g; C) be a scalar triple such that f 2 (X), the convex cone S is closed, and g is S-convex and S-epi closed. Then, the following statements are equivalent:…”

“…This condition is necessary and su¢ cient for the stable Farkas lemma and stable Lagrange duality for (SOP) in [6] (see also [13]). The following example illustrates the ful…lment of (4.5).…”

“…f 2 (X); C is a closed convex set, and g is S-convex and S-epi closed), condition (c 2 ) accounts for the closedness of S z 2S + epi(f + i C + z g) regarding the set f0 X g R (recall that a set A is said to be closed regarding to the set B if B \ cl A = B \ A; see e.g. [2, p. 56]), and this condition is su¢ cient for generalized Farkas lemma for systems involving extended real-valued functions (see, e.g., [6], [13], and [10]). …”