通过非参数学习处理具有决策相关不确定性的上下文随机优化
Contextual Stochastic Optimization with Decision-Dependent Uncertainty via Nonparametric Learning
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中文总结 AI 辅助
研究决策相关上下文随机规划，用非参数回归模型学习不确定性依赖，采用 ER-DD-SAA 框架，开发 MIP 表示，为两阶段问题提出 BD-CG 算法，证明收敛性，建立统计特性，数值实验显示其性能和计算优势。
中文摘要 AI 辅助
我们研究了一种一般的决策相关上下文随机规划（DD-CSP），其中不确定性取决于外生上下文信息和内生决策。为学习不确定性对决策和上下文信息的潜在复杂依赖性，采用了包括 k 近邻（kNN）、分类与回归树（CART）和 ReLU 神经网络等非参数回归模型。考虑预测不确定性时的估计误差，采用基于经验残差的决策相关样本平均近似（ER-DD-SAA）框架。为每个非参数回归模型开发了可无缝嵌入 ER-DD-SAA 框架的精确混合整数规划（MIP）表示。对于具有 kNN 的两阶段 ER-DD-SAA 问题，提出了结合 Bender 分解与约束生成的定制分解算法 BD-CG，并证明其在有限次迭代内收敛到全局最优。从统计角度，在温和正则条件下建立了 ER-DD-SAA 与所有三个非参数回归模型的一致性和渐近最优性。数值实验表明，具有非参数学习的 ER-DD-SAA 模型在样本外性能上始终优于参数基准，所提出的重新表述和算法显著提高了计算可处理性。
英文摘要
We study a general decision-dependent contextual stochastic program (DD-CSP) in which uncertainty depends on both exogenous contextual information and endogenous decisions. To learn the potentially complex dependence of uncertainty on decisions and contextual information, we employ several nonparametric regression models, including k nearest neighbors (kNN), classification and regression trees (CART), and ReLU neural networks. To account for estimation errors in predicting the uncertainty, we adopt an empirical residuals-based decision-dependent sample average approximation (ER-DD-SAA) framework, which adds empirical residuals to the point predictions from the learned regression models. For each nonparametric regression model, we develop exact mixed-integer programming (MIP) representations that can be seamlessly embedded within the ER-DD-SAA framework. For two-stage ER-DD-SAA problems with kNN, we further propose a tailored decomposition algorithm, named BD-CG, that combines Bender's decomposition with constraint generation. Under suitable assumptions, we prove that the proposed BD-CG converges to a global optimum within a finite number of iterations. From a statistical perspective, we establish the consistency and asymptotic optimality of ER-DD-SAA with all three nonparametric regression models under mild regularity conditions. Numerical experiments on a newsvendor problem with pricing and a two-stage facility location problem demonstrate that the ER-DD-SAA model with nonparametric learning consistently outperforms a parametric benchmark in out-of-sample performance and the proposed reformulations and algorithm substantially improve computational tractability.
