融合超度量Gromov-Wasserstein距离用于多阶段随机规划中的场景树生成

中文摘要 AI 辅助

多阶段随机规划需要能够准确近似未知潜在不确定性过程且不违反非预期性约束的场景树。大多数现有的场景树生成方法通常只针对两个目标中的一个，而牺牲另一个，只有一些近期方法试图同时处理这两个目标。在本文中，我们提出将最优传输理论中的两个近期距离，即超度量Gromov-Wasserstein距离和融合Gromov-Wasserstein距离，结合成一个统一的度量，以同时处理这两个目标。我们的主要结果是一个稳定性定理，表明多阶段随机规划问题的最优值变化量受所提出距离在真实过程与其场景树近似之间的Holder型幂次控制，将基于Wasserstein的稳定性结果扩展至同时控制边际保真度和非预期性。我们提出了一种基于所提出距离的块坐标下降算法用于场景树生成，并在不同的库存管理测试案例上进行了评估。

英文摘要

Multistage Stochastic Programming requires scenario trees that accurately approximate the unknown underlying uncertainty process without violating the non-anticipativity constraint. Most of the existing method for scenario tree generation usually target just one of the two goals at the expense of the other, and just some recent methods try to simultaneously address both. In this paper, we propose to combine two recent distances from Optimal Transport Theory, specifically the ultrametric Gromov-Wasserstein and the Fused Gromov-Wasserstein, into a unique measure to simultaneously deal with the two goals. Our main result is a stability theorem showing that the optimal value of a Multistage Stochastic Programming problem changes by an amount controlled by a Holder-type power of the proposed distance between the true process and its scenario tree approximation, extending Wasserstein-based stability results to control marginal fidelity and non-anticipativity, simultaneously. We propose a block coordinate descent algorithm for the our scenario tree generation based on the proposed distance, and evaluate it on different inventory management test cases.

发表机构

• University of Milano-Bicocca（米兰比可卡大学）

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