Connecting Extreme-Point Generation and Decision Rules in Two-Stage Distributionally Robust Optimization

Abstract

Two-stage distributionally robust optimization chooses a here-and-now decision and a wait-and-see decision policy that adapts to uncertainty realizations. Decision-rule methods specify the form of this adaptive policy in advance, whereas decomposition-generation methods construct second-stage value information iteratively. We connect these approaches through extreme points of the second-stage dual problem. Each extreme point defines an affine value piece, and a compatible primal basis can define an affine policy piece. Solving the first-stage problem may require only a subset of these pieces. We specialize an extreme-point generation method to solve the first-stage problem. A separate linear-programming procedure adds pieces until it recovers the recourse value over the uncertainty set, and the pieces can also be used to recover the optimal recourse policy. Building on the algorithm output, we develop a posteriori exactness test for conventional decision rules. We give extensions for degeneracy, rank-deficient recourse, and structured random recourse. The proposed algorithm is computationally efficient on the reported instances, and the results show that completing the second-stage value across the uncertainty set requires more extreme points than solving the first-stage problem alone.