A Unified Approach to Robust Farkas-Type Results with Applications to Robust Optimization Problems

Abstract

In this paper we consider the inequality of the form $ \sup_{u \in \mathcal{U}} F_u(\cdot, 0_Y) \geq h,$ where $X, Y$ are locally convex Hausdorff topological vector spaces, $\mathcal{U} \not= \emptyset$ is an uncertainty set, $F_u : X \times Y \rightarrow \overline{\mathbb{R}}$ for each $u \in \mathcal{U}$, and $h : X \rightarrow \overline{\mathbb{R}}$ is a lower semicontinuous proper convex function. Characterizations of such an inequality in terms of robust abstract perturbational duality are established and applied to diverse robust composite functional inequalities to obtain variants of robust Farkas-type results, such as robust Farkas lemmas for general nonconvex conical systems, robust Farkas lemmas for convex-DC systems, robust/stable robust Farkas lemmas for convex systems, robust Farkas lemmas for general linear systems in infinite dimensional spaces, and robust semi-infinite Farkas lemmas. The results are then applied to classes of robust DC and robust convex optimization problems, and strong F...