Prescriptive PCA: Dimensionality Reduction for Two-stage Stochastic
  Optimization
Long He Ho-Yin Mak
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.