两阶段样本鲁棒优化何时渐近最优？一个简单视角
When Is Two-Stage Sample Robust Optimization Asymptotically Optimal? A Simple Perspective
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中文总结 AI 辅助
本文通过支撑集的单纯性特征解决了两阶段样本鲁棒优化渐近最优性的开放问题，明确了其成立与否的二分条件，并提出了验证技术性条件的多项式时间算法。
中文摘要 AI 辅助
带线性决策规则的两阶段样本鲁棒优化是针对分布未知的两阶段随机线性规划的标准数据驱动方法。Bertsimas等人（2022）证实该方法在若干条件下渐近最优，其中多数条件较宽松，但存在一个技术性条件，且该条件是否可由其他条件推导是待解决的开放问题。本文通过支撑集的单一几何特征——单纯性（即每个顶点关联的边数恰好等于环境维度的性质）解决该开放问题：当支撑集单纯时，该技术性条件自动满足，在其余标准假设下渐近最优成立；当支撑集不单纯时，存在技术性条件不满足的问题实例，渐近最优性失效。这一结果以否定形式解决了开放问题，形成了明确的二分性，且具有直接实际意义，因为广泛使用的支撑集可能不单纯。针对此类支撑集，本文提供了一个多项式时间算法，仅利用顶点关联边的方向即可在任意给定顶点处验证该技术性条件，案例研究表明该算法有效。
英文摘要
Two-stage sample robust optimization with linear decision rules is a standard data-driven approach to two-stage stochastic linear programs with unknown distributions. \cite{bertsimas2022two} established that this approach is asymptotically optimal under several conditions. Most are mild, but one is technical, and whether it follows from the others was left as an open question. We resolve this open question through a single geometric feature of the support set: simpleness, the property that every vertex is incident to exactly as many edges as the ambient dimension. When the support set is simple, the technical condition is automatic, and asymptotic optimality holds under the remaining standard assumptions. When the support set is not simple, there exists a problem instance on which the technical condition fails, and asymptotic optimality breaks down. This settles the open question in the negative and yields a sharp dichotomy. The dichotomy has direct practical consequences, since widely used support sets may not be simple. For such supports, we provide a polynomial-time algorithm that certifies the technical condition at any given vertex using only its incident edge directions. Our case study illustrates the effectiveness of our algorithm.
