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The original motivation for this paper was to provide an efficientquantitative analysis of convex infinite (or semi-infinite) inequality systemswhose decision variables run over general infinite-dimensional (resp.finite-dimensional) Banach spaces and that are indexed by an arbitrary fixedset $J$. Parameter perturbations on the right-hand side of the inequalities arerequired to be merely bounded, and thus the natural parameter space is$l_{\infty}(J)$. Our basic strategy consists of linearizing the parameterizedconvex system via splitting convex inequalities into linear ones by using theFenchel-Legendre conjugate. This approach yields that arbitrary boundedright-hand side perturbations of the convex system turn on constant-by-blocksperturbations in the linearized system. Based on advanced variational analysis,we derive a precise formula for computing the exact Lipschitzian bound of thefeasible solution map of block-perturbed linear systems, which involves onlythe system's data, and then show that this exact bound agrees with thecoderivative norm of the aforementioned mapping. In this way we extend to theconvex setting the results of [3] developed for arbitrary perturbations with noblock structure in the linear framework under the boundedness assumption on thesystem's coefficients. The latter boundedness assumption is removed in thispaper when the decision space is reflexive. The last section provides the aimedapplication to the convex case.

Quantitative Stability of Linear Infinite Inequality Systems under Block Perturbations with Applications to Convex Systems