两阶段分布鲁棒优化中极值点生成与决策规则的关联
Connecting Extreme-Point Generation and Decision Rules in Two-Stage Distributionally Robust Optimization
AI总结：
本研究关联两阶段分布鲁棒优化的极值点生成与决策规则方法，提出专门化算法及后验精确性测试，验证其计算高效性。
AI中文摘要：
两阶段分布鲁棒优化会选择一个“即刻执行”决策和一个适应不确定性实现情况的“观望”决策策略。决策规则方法预先指定该自适应策略的形式，而分解-生成方法则迭代构建第二阶段的价值信息。我们通过第二阶段对偶问题的极值点将这些方法关联起来：每个极值点定义一个仿射价值片段，兼容的原始基可定义一个仿射策略片段，求解第一阶段问题可能仅需这些片段的子集。我们将极值点生成方法专门化以求解第一阶段问题，一个独立的线性规划程序会添加片段，直至覆盖不确定性集上的追索价值，这些片段还可用于恢复最优追索策略。基于算法输出，我们为传统决策规则开发了后验精确性测试，并针对退化、秩亏追索及结构化随机追索给出了扩展方案。所提算法在报告的实例上计算高效，结果表明，要完整覆盖不确定性集上的第二阶段价值，所需的极值点数量多于仅求解第一阶段问题所需的数量。
英文摘要：
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.
