Asymptotic Farkas lemmas for convex systems

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 Kconvex 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


ASTRACT
In this paper we establish characterizations of the containment of the set { : , ( ) 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 Keywords: Farkas lemma, sequential Farkas lemma, limit inferior, limit superior

INTRODUCTION AND PRELIMINARIES
Farkas-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 (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).In this paper, we mainly established several forms of asymptotic Farkas-type results for convex systems in the two means: systems convex with respect to a convex cone (called -convex systems) and systems convex w.r.t. an extended sublinear function S (called S -convex systems).The results can be used to establish the optimality conditions and dulaity for optimization problems where constraint qualification conditions failed, such as classes of semidefinite programs, or scalarized multi-objective programs, scalarized vector optimization problems.We shoned also that under some closedness conditions, these asymptotic Farkas lemmas came back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature.
Let X and Y be lcHtvs, with their topological dual spaces X  and Y  , respectively.The only topology we consider on The set of all proper, lower semi-continuous (lsc in short) and convex functions on X will be denoted by  .
where " K  " is the binary relation (generated by We consider also the extension of S as :   → ℝ ∪ {+∞} By setting Given an extended sublinear function :  → ℝ ∪ {+∞} , we adapt the notion S  convex ( i.e., convex with respect to a sublinear function) in [6] which generalized the one in [16].

It is clear that
h is K -convex if and only if epi K h is convex.In addition, : "affine", "convex", "concave" or "arbitrary", respectively.Moreover, the equalities hold whenever one of the nets is convergent.[6]).Definition 1.1 [2, p.5] [1, p.32], [14, p.217] Let () a  be a net of extended real numbers defined on a directed set ( Ι, ≫) e define limit inferior of the net () Similarly, limit superior of the net () provide that the right side of the inequalities are defined.

Approximate Farkas lemma for cone-convex systems
In this section we will establish one of the main result of this paper: the asymptotic version of Farkas lemma for convex systems, which holds without any qualification condition.

Let ,
XY lcHtvs, K be a closed convex cone in Y , C be a nonempty closed convex subset of X and :  → ℝ ∪ {+∞} be a proper lsc and convex function.Let further : The following statements are equivalent: x   (iv) there exists a net * () A X fi  Since we also have [4, p. 328 U and the equivalence between (i) and (ii) follows.
Trang 163 By the definition of the conjugate function, (2.1) implies that x  for all iI  .Then the above inequality gives rise to ). Taking liminf in both sides of the last inequality, we get (iv).[(iv) (i)]  Assume that (iv) holds, i.e., there exists a net * () was established in [5] and the following statements: As shown in the proof of Theorem 2.1, (i) is equivalent to * * (0 , 0) epi( ) .
Therefore, one also gets (b).The proof is complete.
Corollary 2.1 [Farkas lemma for cone-convex systems] Consider the following conditions: and the following statements: As in the proof of Theorem 2.1, one has (i) is equivalent to * (0 ,0) cl .

X  D
Moreover, it is easy to check that (v) is equivalent to * (0 , 0) X D.Thus we get (a).
Therefore, one also gets (b).The proof is complete.
and dom
Trang 166 Passing to the limit both sides of the last inequality and taking the fact that (3.4) into account, we get (a).The proof is complete.