两阶段随机混合整数规划的场景约简
Scenario Reduction for Two-Stage Stochastic Mixed-Integer Programs
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中文总结 AI 辅助
研究两阶段随机混合整数规划的场景约简问题，基于概率分布距离和最优质量运输问题提出新成本函数，用前向选择算法证明其优势，还提出混合算法，通过案例研究表明新函数和算法能有效提升求解质量与效率。
中文摘要 AI 辅助
两阶段随机混合整数规划是不确定性下决策的重要工具，但用大量场景表示不确定性并确保准确结果会使其求解具有挑战性。场景约简通过找到支持在较少场景上且仍产生相似最优第一阶段决策的分布来解决此问题。本文重新审视基于概率分布间距离和最优质量运输问题的经典场景约简理论，回顾并比较文献中的各种运输成本函数并提出新函数。使用前向选择算法证明新成本函数能从给定样本中首次抽取时就以相对近似误差选出最佳场景。为降低评估成本函数的计算成本，进一步提出包含场景预选择阶段的混合算法。在小型24节点和大型300节点案例研究的两阶段随机机组组合问题上评估了解决方案质量和计算复杂度。仅约五个场景时，新成本函数对小型和大型案例分别将全分布最优近似到约2.1%和0.4%的误差内。混合算法在大型案例研究中实现了相似的解决方案质量，同时将挂钟时间减少了18倍，工作量（由Gurobi求解器衡量）减少了66倍。
英文摘要
Two-stage stochastic mixed-integer programs are important tools for decision-making under uncertainty. Representing the uncertainty with many scenarios, however, can make them challenging to solve. Scenario reduction addresses this by finding a distribution supported on fewer scenarios that still yields similar optimal first-stage decisions. In this paper, we revisit the classical scenario reduction theory based on distances between probability distributions and the optimal mass transportation problem. The transportation problem's cost function captures scenario similarity and is central to the effectiveness of scenario reduction. We then review and compare various transportation cost functions from the literature and propose a new one. Using the Forward Selection Algorithm, we prove that our proposed cost function selects the best possible scenario from a given sample on the first draw with respect to the relative approximation error. To reduce the computational cost of evaluating this cost function, we further propose a hybrid algorithm with a scenario pre-selection phase. We assess solution quality and computational complexity on the two-stage stochastic unit commitment problem for small 24-bus and large 300-bus case studies. With only around five scenarios, the proposed cost function approximates the full-distribution optimum to within roughly 2.1% and 0.4% error for the small and large cases, respectively. In contrast, prevalent cost functions often need 25 scenarios or more to achieve that solution quality. The hybrid algorithm achieves similar solution quality while reducing wall-clock time by a factor of 18 and work (per Gurobi solver) by a factor of 66 on the large case study.
