Robustness Measures in Distributionally Robust Optimization
For practitioners and theorists in optimization and decision-making under uncertainty, this work provides a principled framework for choosing uncertainty sets and understanding the trade-off between performance and robustness.
The paper shows that the regularizer in distributionally robust optimization (DRO) is a worst-case sensitivity measure, enabling a systematic approach to selecting uncertainty sets and tracing a near Pareto-optimal performance-robustness frontier. This reveals when the price of robustness is high and how to redesign systems to reduce it.
Distributionally Robust Optimization (DRO) is a worst-case approach to decision making when there is model uncertainty. It is also well known that for certain uncertainty sets, DRO is approximated by a regularized nominal problem. We show that the regularizer is not just a penalty function but the worst-case sensitivity (WCS) of the expected cost with respect to deviations from the nominal model, giving it the interpretation of a robustness measure. This has substantial consequences for robust modeling. It shows that DRO is fundamentally a tradeoff between performance and robustness, where the robustness measure is determined by the uncertainty set. The robustness measure reveals properties of a cost distribution that affect sensitivity to misspecification. This leads to a systematic approach to selecting uncertainty sets. The family of DRO solutions obtained by varying the size of the uncertainty set traces a near Pareto-optimal performance--robustness frontier that can be used to select its size. The frontier identifies problem instances where the price of robustness is high and provides insight into effective ways of redesigning the system to reduce this cost. We derive WCS for a collection of commonly used uncertainty sets, and illustrate these ideas in a number of applications.