Costs and benefits of robust optimization

In this exposition the robust counterpart approach by Ben-Tal, El Ghaoui and Nemirovski is investigated with respect to its costs and benefits, with the focus on the costs of robustification. Although robust optimization has gained more and more interest among both academics and practitioners and although this certainly represents a well-established theory, it is to some extent unclear, if and what costs have to be beared when using the robust counterpart formulation. Further, it is not known if other benefits besides the obvious can be realized by robustification. Concerning benefits, there is only one theoretical result by El Ghaoui and Lebret in the special situation of robust least squares besides the obvious robustness in feasibility under perturbations. In addition, on the cost side, one of the earlier papers [3] by Ben-Tal and Nemirovski provides a stability analysis together with a result concerning costs for robust linear optimization under convex uncertainty. In the following, Ben-Tal and Nemirovski’s results on costs are generalized to smooth convex conic problems under Lipschitz uncertainty, given reasonably mild regularity conditions. For robust linear optimization, it is shown that under affine uncertainty and ellipsoidal uncertainty set, uniqueness of the optimal robust solution may be achieved as additional benefit in most situations.