Consistency of robust optimization with application to portfolio optimization

In recent years the robust counterpart approach, introduced and made popular by Ben-Tal, Nemirovski and El Ghaoui, gained more and more interest among both academics and practitioners. However, to the best of our knowledge, only very few results on the relationship between the original problem instance and the robust counterpart have been established. This exposition aims at closing this gap by showing that the robust counterpart to an already well-posed problem remains well-posed under some mild regularity and uniqueness assumption on the solution of the original problem instance. As a consequence, sufficient conditions will be established under which the solution of the robust counterpart converges to the original solution, if the level of robustification is decreased to zero. Based on the well-posedness of the robust counterpart, it will also be demonstrated how any consistent plug-in estimator can be supplemented by a corresponding consistent robust estimator based on a proper choice of the confidence set of the plug-in estimator. Finally, this consistency result leads to a generalization of already known consistency results in the framework of mean-variance portfolio optimization.