Duality for Robust Linear Infinite Programming Problems Revisited

• Published in: Vietnam journal of mathematics Vol. 48; no. 3; pp. 589 - 613
• Main Authors: Dinh, N., Long, D. H., Yao, J.-C.
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.09.2020; Springer Nature B.V
• Subjects: Linear systems; Mathematics; Mathematics and Statistics; Original Article; Robustness; Uncertainty; 90C31; 39B62; Stable robust strong duality for robust linear infinite problems; Robust Farkas-type results for sub-affine systems; Linear infinite programming problems; 49J52; 90C25; Robust linear infinite problems; 46N10; Robust Farkas-type results for infinite linear systems
• ISSN: 2305-221X; 2305-2228
• DOI: 10.1007/s10013-020-00383-6

In this paper, we consider the robust linear infinite programming problem (RLIP c ) defined by ( RLIP c ) inf 〈 c , x 〉 subject to x ∈ X , 〈 x ∗ , x 〉 ≤ r , ∀ ( x ∗ , r ) ∈ U t , ∀ t ∈ T , where X is a locally convex Hausdorff topological vector space, T is an arbitrary index set, c ∈ X ∗ , and U t ⊂ X ∗ × ℝ , t ∈ T are uncertainty sets. We propose an approach to duality for the robust linear problems with convex constraints (RP c ) 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 (RLIP c ), one gets back to the model (RP c ) and the (stable) robust strong duality for (RP c ) applies. By such a way, nine versions of dual problems for (RLIP c ) 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., (RLIP c ) 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.