New Farkas-Type Results for Vector-Valued Functions: A Non-abstract Approach

Abstract

This paper provides new Farkas-type results characterizing the inclusion A ⊂ B, where A and B are subsets of a locally convex space X. The sets A and B are described here by means of vector functions from X to other locally convex spaces Y (equipped with the partial ordering associated with a given convex cone K ⊂ Y) and Z, respectively. These new Farkas lemmas are obtained via the complete characterization of the K-epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the autors [J. Optim. Theory Appl. 173, 357-390 (2017)], the characterizations of A ⊂ B are expressed here in terms of the data.

Sectional Convexity of Epigraphs of Conjugate Mappings with Applications to Robust Vector Duality

“…In Z • we adopt the same conventions as in (2.6). Moreover, we recall the cone of positive operators (see [1], [8]) and the cone of weak positive operators [11] respectively, as follows:…”

“…Our purpose is to generalize the representation in Lemma 4.1 to the vector case. The difficulty in such a generalization is that the set epi(F + I A ) * in general is not convex [11,Example 2.6], and hence, it is almost no hope for a representation of the same form as in (4.9). Fortunately, with the help of Proposition 3.3, (4.9) can be generalized with the use of the k-sectionally convex hull, as shown in the next theorem.…”

“…Also, F −L is a K-convex mapping. Thus, (F −L)(A ∩ dom F ) + int K is convex (see [11,Remark 4.1]), and so is (L − F )(A ∩ dom F ) − int K.…”

“…In such a case (U is a singleton) Proposition 3.3 givesÃk ⊂à ⊂B ⊂ epi(F + I B ) * , which, together with the fact that epi(F + I B ) * is closed (see [11,Lemma 3.6]), leads to clÃk ⊂ clà ⊂ clB ⊂ epi(F + I B ) * .…”

“…Note that in the first part of the proof of[11, Theorem 4.2] no assumptions on the convexity or closedness of the mappings F and G are needed.…”