  We consider the stability of Robust Optimization problems with respect to
perturbations in their uncertainty sets. We focus on Linear Optimization
problems, including those with a possibly infinite number of constraints, also
known as Linear Semi-Infinite Optimization (LSIO) problems, and consider
uncertainty in both the cost function and constraints. We prove Lipschitz
continuity of the optimal value and {\epsilon}-approximate optimal solution set
with respect to the Hausdorff distance between uncertainty sets and with an
explicit Lipschitz constant that can be calculated. In addition, we prove
closedness and upper semi-continuity for the optimal solution set mapping with
respect to the uncertainty set.
