Abstract
In this paper, we proved a new extended version of the Hahn-Banach-Lagrange theorem that is valid in the absence of a qualification condition and is called an approximate Hahn-Banach-Lagrange theorem. This result, in special cases, gives rise to approximate sandwich and approximate Hahn-Banach theorems. These results extend the Hahn-Banach-Lagrange theorem, the sandwich theorem in [18], and the celebrated Hahn-Banach theorem. The mentioned results extend the original ones into two features: Firstly, they extend the original versions to the case with extended sublinear functions (i.e., the sublinear functions that possibly possess extended real values). Secondly, they are topological versions which held without any qualification condition. Next, we showed that our approximate Hahn-Banach-Lagrange theorem was actually equivalent to the asymptotic Farkas-type results that were established recently [10]. This result, together with the results [5, 16], give us a general picture on the equivalence of the Farkas lemma and the Hahn-Banach theorem, from the original version to their corresponding extensions and in either non-asymptotic or asymptotic forms.
Highlights
In this paper, we proved a new extended version of the Hahn-Banach-Lagrange theorem that is valid in the absence of a qualification condition and is called an approximate HahnBanach-Lagrange theorem
We will recall the sequential Farkas lemmas for convex systems in [10] which hold without any qualification condition: the asymptotic version of the Farkas lemma for systems which is convex w.r.t. a convex cone and the one for systems which is convex w.r.t. an extended sublinear function
The conclusion follows from Theorem 3.1 by taking Y X, C : X and g(x) : x for all Concretely, we show that two versions of sequential Farkas lemma for cone-convex systems and for sublinear-convex systems in [10] and the approximate Hahn-Banach-Lagrange established in this paper are equivalent
Summary
We will recall the sequential Farkas lemmas for convex systems in [10] which hold without any qualification condition: the asymptotic version of the Farkas lemma for systems which is convex w.r.t. a convex cone and the one for systems which is convex w.r.t. an extended sublinear function. Let X, Y be lcHtvs, K be a closed convex cone in Y , C be a nonempty closed convex subset of X and f : X ¡ { } be a proper lsc and convex function
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