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

This paper concerns the robust vector problem $$ \mathrm{(RVP)}\enskip \text{WMin}\left\{ F(x) : x\in C, G_{u}(x)\in -S,\forall u\in\mathcal{U}\right\}, $$ where X,Y,Z are locally convex Hausdorff topological vector spaces, K is a closed and convex cone in Y with a nonempty interior, and S is a closed, convex cone in Z, $\mathcal {U}$ is an uncertainty set, $F\colon X\rightarrow {Y}^{\bullet },$$G_{u}\colon X\rightarrow Z^{\bullet }$ are proper mappings for all $ u \in \mathcal U$, and ∅ ≠ C ⊂ X. Let $ A:=C\cap \left (\bigcap _{u\in \mathcal {U}}G_{u}^{-1}(-S)\right )$ and IA : X → Y∙ be the indicator map defined by IA(x) = 0Y if x ∈ A and $I_{A}(x) = + \infty _{Y}$ if x∉A. It is well known that the epigraph of the conjugate mapping (F + IA)∗, in general, is not a convex set. We show that, however, it is “k-sectionally convex” in the sense that each section formed by the intersection of epi(F + IA)∗ and any translation of a “specific k-direction-subspace” is a convex subset, for any k taking from intK. The key results of the paper are the representations of the epigraph of the conjugate mapping (F + IA)∗ via the closure of the k-sectionally convex hull of a union of epigraphs of conjugate mappings of mappings from a family involving the data of the problem (RVP). The results then give rise to stable robust vector/convex vector Farkas lemmas which, in turn, are used to establish new results on robust strong stable duality results for (RVP). It is shown at the end of the paper that, when specifying the result to some concrete classes of scalar robust problems (i.e., when $Y = \mathbb R$), our results cover and extend several corresponding known ones in the literature.