Stability and Continuity in Robust Optimization
研究了鲁棒优化问题中不确定集扰动对最优值和近似最优解集的影响，证明了其关于豪斯多夫距离的利普希茨连续性，并给出了利普希茨常数的计算方法。
We consider the stability of robust optimization (RO) problems with respect to 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 constraints. We prove Lipschitz continuity of the optimal value and $\epsilon$-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 (LSIO) 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.