The Stability of Robustness for Conic Linear Programs with Uncertain Data

The robust counterpart of a given conic linear program with uncertain data in the constraints is defined as the robust conic linear program that arises from replacing the nominal feasible set by the robust feasible set of points that remain feasible for any possible perturbation of the data within an uncertainty set. Any minor changes in the size of the uncertainty set can result in significant changes, for instance, in the robust feasible set, robust optimal value and the robust optimal set. The concept of quantifying the extent of these deviations is referred to as the stability of robustness. This paper establishes conditions for the stability of robustness under which minor changes in the size of the uncertainty sets lead to only minor changes in the robust feasible set of a given linear program with cone constraints and ball uncertainty sets.