UncertaintySets

Uncertainty sets are mathematical representations of the possible values that uncertain parameters may take in optimization and decision problems. They encode available information, such as historical data, expert judgment, or statistical bounds, and serve as the foundation for robust decision making. In robust optimization, decisions are chosen to perform well for all realizations of the uncertain parameters within the specified set.

Common uncertainty set shapes include boxes (intervals for each parameter), ellipsoids, and polyhedra. Box sets are simple and lead to linear reformulations, while ellipsoidal and other convex sets can yield second-order cone or semidefinite program formulations. Polyhedral sets, including budgeted or sum-constrained sets, balance tractability with expressive power. Convexity is often assumed to maintain computational tractability, though non-convex sets are also studied.

The robust counterpart of an optimization problem typically replaces uncertain coefficients by their worst-case values over the uncertainty set. This leads to solutions that guarantee feasibility under all parameter realizations within the set. The choice of set directly affects tractability, conservatism, and performance: larger sets provide greater protection but can be overly conservative, whereas smaller sets may be more efficient but risk infeasibility if real-world values lie outside the set.

Uncertainty sets are frequently data-driven, calibrated from historical observations, confidence bounds, or estimated variances. They appear across fields such as supply chain, finance, energy systems, and control. Extensions include dynamic or multistage uncertainty sets for problems with evolving information over time, as well as scenario-based or adaptive formulations.