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

TL;DR

This paper develops a precise quantitative analysis of the stability of infinite convex inequality systems under block-structured perturbations, extending previous linear results to the convex setting using advanced variational analysis.

Contribution

It introduces a method to compute the exact Lipschitzian bound of the feasible solution map for convex systems with block perturbations, generalizing prior linear results to convex systems.

Findings

Derived a formula for the Lipschitzian bound involving system data

Extended stability results from linear to convex systems

Removed boundedness assumption in reflexive spaces

Abstract

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set $J$. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is $l_{\infty}(J)$. Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of [3] developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.