A major bottleneck in scenario-based Sample Average Approximation (SAA) for stochastic programming (SP) is the cost of solving an exact second-stage problem for every scenario, especially when each scenario contains an NP-hard combinatorial structure. This has led much of the SP literature to restrict the second stage to linear or simplified models. We develop a GPU-based framework that makes full-fidelity integer second-stage models tractable at scale. The key innovation is a set of hardware-aware, scenario-batched GPU kernels that expose parallelism across scenarios, dynamic-programming (DP) layers, and route or action options, enabling Bellman updates to be executed in a single pass over more than 1,000,000 realizations. We evaluate the approach in two representative SP settings: a vectorized split operator for stochastic vehicle routing and a DP for inventory reinsertion. Implementation scales nearly linearly in the number of scenarios and achieves a one-two to four-five orders of magnitude speedup, allowing far larger scenario sets and reliably stronger first-stage decisions. The computational leverage directly improves decision quality: much larger scenario sets and many more first-stage candidates can be evaluated within fixed time budgets, consistently yielding stronger SAA solutions. Our results show that full-fidelity integer second-stage models are tractable at scales previously considered impossible, providing a practical path to large-scale, realistic stochastic discrete optimization.

翻译：基于场景的样本平均逼近（SAA）方法在随机规划（SP）中的一个主要瓶颈在于需为每个场景精确求解第二阶段问题所带来的计算成本，尤其当每个场景包含NP难组合结构时。这导致大量SP文献将第二阶段限制为线性或简化模型。我们开发了一个基于GPU的框架，使得完整保真度的整数第二阶段模型在大规模场景下可高效求解。核心创新在于一组硬件感知、场景批处理的GPU内核，该设计充分挖掘了跨场景、动态规划（DP）层级以及路径或行动选项的并行性，使得贝尔曼更新可在单次遍历中处理超过1,000,000个实现。我们在两个典型SP场景中评估了该方法：用于随机车辆路径规划的向量化分割算子，以及用于库存再插入的动态规划。该实现方案在场景数量上呈现近似线性扩展，并实现二至四个数量级到四至五个数量级的加速，从而支持远大于以往的场景集和更可靠的第一阶段决策。计算优势直接提升了决策质量：在固定时间预算内可评估更大规模的场景集和更多的第一阶段候选方案，持续产生更强的SAA解。我们的结果表明，完整保真度的整数第二阶段模型在以往认为不可行的规模下是可处理的，为大规模、现实的随机离散优化问题提供了可行的解决路径。