Duality for Robust Linear Infinite Programming Problems Revisited

In this paper, we consider the robust linear infinite programming problem (RLIPc) defined by (RLIPc)inf〈c,x〉subject tox∈X,〈x∗,x〉≤r,∀(x∗,r)∈Ut,∀t∈T,where Xis a locally convex Hausdorff topological vector space, Tis an arbitrary index set, c∈ X∗, and Ut⊂X∗×ℝ, t∈ Tare uncertainty sets. We propose an approach to duality for the robust linear problems with convex constraints (RPc) and establish corresponding robust strong duality and also, stable robust strong duality, i.e., robust strong duality holds “uniformly” with all c∈ X∗. With the different choices/ways of setting/arranging data from (RLIPc), one gets back to the model (RPc) and the (stable) robust strong duality for (RPc) applies. By such a way, nine versions of dual problems for (RLIPc) are proposed. Necessary and sufficient conditions for stable robust strong duality of these pairs of primal-dual problems are given, for which some cover several known results in the literature while the others, due to the best knowledge of the authors, are new. Moreover, as by-products, we obtained from the robust strong duality for variants pairs of primal-dual problems, several robust Farkas-type results for linear infinite systems with uncertainty. Lastly, as extensions/applications, we extend/apply the results obtained to robust linear problems with sub-affine constraints, and to linear infinite problems (i.e., (RLIPc) with the absence of uncertainty). It is worth noticing even for these cases, we are able to derive new results on (robust/stable robust) duality for the mentioned classes of problems and new robust Farkas-type results for sub-linear systems, and also for linear infinite systems in the absence of uncertainty.