A unified approach to inverse robust optimization problems

Abstract: A variety of approaches has been developed to deal with uncertain optimization problems. Often, they start with a given set of uncertainties and then try to minimize the influence of these uncertainties. Depending on the approach used, the corresponding price of robustness is different. The reverse view is to first set a budget for the price one is willing to pay and then find the most robust solution. In this article, we aim to unify these inverse approaches to robustness. We provide a general problem definition and a proof of the existence of its solution. We study properties of this solution such as closedness, convexity, and boundedness. We also provide a comparison with existing robustness concepts such as the stability radius, the resilience radius, and the robust feasibility radius. We show that the general definition unifies these approaches. We conclude with examples that demonstrate the flexibility of the introduced concept.