Farkas Lemma for Convex Systems Revisited and Applications to Sublinear-Convex Optimization Problems

An approximate Hahn-Banach-Lagrange theorem

“…The generalization of these reults in comparison with [5,18] is twofold: firstly, they extend the original version to the case with extended sublinear functions (i.e., the sublinear functions which possibly possess extended real values); secondly, Science & Technology Development, Vol 19, No.T6-2016 Trang 170 in contrast to [5], they are topological versions which hold without any qualification condition. The paper can be considered as a continuation of the previous ones (of the authors and their coauthors) [5,10,12]. Some tools and some ideas of generalizations to Hahn-Banach-Lagrange theorem and to real-extended sublinear functions are modifications of the one in [5] to adapt to the case where no qualification condition is assumed.…”

Asymptotic Farkas lemmas for convex systems

“…Typical Farkas lemma for cone-convex systems characterizes the containment of the set where is a closed convex subset of a locally convex Hausdorff topological vector space (brieftly, lcHtvs), is a closed convex cone in another lcHtvs and is aconvex mapping, in a reverse convex set, define by the proper, lower semi-continuous, convex function. If the characterization holds under some constraint qualification condition or qualification condition then it is called non-asymptotic Farkas-type result (see [6], [10][11][12]). Otherwise (i.e., without any qualification condition), such characterizations often hold in the limit forms and they are called asymptotic Farkas-type results (see [7,5,9,13] and references therein).…”