A Consistency-Robustness Framework for Robust Optimization: Integrating Predictions into Robust Scheduling
Robust optimization protects against uncertainty by optimizing for the worst case over a prescribed uncertainty set. This protection can be overly conservative when forecasts, historical data, or learned predictions indicate a more likely scenario. We introduce a framework for robust optimization with predictions. The input consists of an uncertainty set together with a distinguished predicted scenario, and the goal is to compute a single solution that is both consistent, meaning near-optimal for the predicted scenario, and robust, meaning competitive with the classical min-max robust optimum. Unlike in standard learning-augmented algorithms, the prediction does not merely estimate the realized input; it creates a separate benchmark, the predicted optimum, which must be balanced against the min-max robust optimum. We study this framework for makespan scheduling with uncertain processing times and give a structural classification across standard uncertainty models and machine environments. For interval uncertainty, we obtain a smooth $(1+1/\lambda,1+\lambda)$ consistency-robustness tradeoff for restricted-assignment and related machines. Furthermore, we prove that unrelated machines admit no constant tradeoff. For budgeted uncertainty, we obtain a $(1+1/\lambda,2+\lambda)$ tradeoff for restricted assignment. Our analysis is based on a duality-based reduction to an interval-like upper envelope. We complement this with a lower bound showing that related machines admit no constant tradeoff even when only one job may deviate. For arbitrary uncertainty sets, we obtain constant tradeoffs for identical machines via a support-function block construction, and prove impossibility for restricted assignment. Our results show that the possibility of combining consistency and robustness in robust scheduling depends critically on the interaction between the uncertainty model and the machine environment.