Duality for Robust Linear Infinite Programming Problems Revisited

TL;DR

This paper develops a duality framework for robust linear infinite programming problems, establishing strong duality conditions, exploring multiple dual problem variants, and deriving new robust Farkas-type results with applications to linear problems.

Contribution

It introduces nine dual problem variants for robust linear infinite programming and provides necessary and sufficient conditions for their stable strong duality, including new theoretical results.

Findings

Nine dual problem variants proposed for robust linear infinite programming.

Necessary and sufficient conditions established for stable robust strong duality.

New robust Farkas-type results derived from duality analysis.

Abstract

In this paper, we consider the robust linear infinite programming problem $({\rm RLIP}_c) $ defined by \begin{eqnarray*} ({\rm RLIP}_c)\quad &&\inf\; \langle c,x\rangle \textrm{subject to } &&x\in X,\; \langle x^\ast,x \rangle \le r ,\;\forall (x^\ast,r)\in\mathcal{U}_t,\; \forall t\in T, \end{eqnarray*} where $X$ is a locally convex Hausdorff topological vector space, $T$ is an arbitrary (possible infinite) index set, $c\in X^*$, and $\mathcal{U}_t\subset X^*\times \mathbb{R}$, $t \in T$ are uncertainty sets. We propose an approach to duality for the robust linear problems with convex constraints $({\rm RP}_c)$ and establish corresponding robust strong duality and also, stable robust strong duality, With the different ways of arranging data from $({\rm RLIP}_c) $, one gets back to the model $({\rm RP}_c)$ and the (stable) robust strong duality for $({\rm RP}_c)$ applies. By such a way, nine versions of dual problems for $ ({\rm RLIP}_c)$ are proposed. Necessary and sufficient conditions for stable robust strong duality of these pairs of primal-dual problems are given, 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 applications, we get the results for robust linear problems with sub-affine constraints, and to linear infinite problems (i.e., $({\rm RLIP}_c) $ with the absence of uncertainty).