STABILITY AND CONTINUITY IN ROBUST OPTIMIZATION

We consider the stability of robust optimization (RO) problems with respect perturbations in their uncertainty sets. In particular, We focus on robust linear optimization problems, including those with an infinite number of constraints, and consider uncertainty in both the cost function and coast rants. We prove Lipschitz continuity of the optimal value and is an element of-approximate optimal solution set with respect to the Hausdorff distance between uncertainty sets. The Lipschitz constant can be calculated from the problem data. In addition, we prove closedness and upper semicontinuity for the optimal solution set with respect) to the uncertainty set. In order to prove these results, we develop a novel transformation that maps RO problems to linear semi-infinite optimization (LISO) problems in such a way that the distance between uncertainty sets of two RO problems correspond to a measure of distance between their equivalent LSIO problems. Using this isometry we leverage LSIO and variational analysis stability results to obtain stability results for RO problems.