Sectional convexity of epigraphs of conjugate mappings with applications to robust vector duality

Abstract

This paper concerns the robust vector problem (RVP) W Min { F(x) : x € C. G„ (x) є -S, V« є U. 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, U is an uncertainty set, F : X —> Y*, G„ : X —> Z* are proper mappings for all u € U. and θ ≠ C ⊂ X. 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 = R), our results cover and extend several corresponding known ones in the literature.