In this paper, we consider the alignment between an upstream dimensionality
reduction task of learning a low-dimensional representation of a set of
high-dimensional data and a downstream optimization task of solving a
stochastic program parameterized by said representation. In this case, standard
dimensionality reduction methods (e.g., principal component analysis) may not
perform well, as they aim to maximize the amount of information retained in the
representation and do not generally reflect the importance of such information
in the downstream optimization problem. To address this problem, we develop a
prescriptive dimensionality reduction framework that aims to minimize the
degree of suboptimality in the optimization phase. For the case where the
downstream stochastic optimization problem has an expected value objective, we
show that prescriptive dimensionality reduction can be performed via solving a
distributionally-robust optimization problem, which admits a semidefinite
programming relaxation. Computational experiments based on a warehouse
transshipment problem and a vehicle repositioning problem show that our
approach significantly outperforms principal component analysis with real and
synthetic data sets.

本文讨论了将高维数据的低维表示学习任务与用该表示解决随机规划的优化任务进行对齐，提出了通过解决分布鲁棒优化问题实现预制降维的框架。实验表明，该方法在仓库中转问题和车辆重定位问题上明显优于传统的主成分分析方法。