Abstract
In this paper we establish characterizations of the containment of the set {xX: xC,g(x)K}{xX: f (x)0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g:X Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. Here, no constraint qualification condition or qualification condition are assumed. The characterizations are often called asymptotic Farkas-type results. The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function. It is also shown that under some closedness conditions, these asymptotic Farkas lemmas go back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature. The results can be used to study the optimization
Highlights
AND PRELIMINARIESFarkas-type results have been used as one of the main tools in the theory of optimization [8]
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, is a closed convex cone in another lcHtvs and is a convex 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 it is called non-asymptotic Farkas-type result
Summary
Tien Giang University, Tien Giang (Received on June 5th 2015, accepted on November 21th 2016). {x X: x C, g(x) K} {x X: f (x) 0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g : X Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. No constraint qualification condition or qualification condition are assumed. The characterizations are often called asymptotic Farkas-type results. The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function. The results can be used to study the optimization
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