Prescriptive PCA: Dimensionality Reduction for Two-stage Stochastic Optimization
This addresses the problem of suboptimal decision-making in two-stage stochastic optimization for operations research domains, offering a novel integration of dimensionality reduction and optimization.
The paper tackles the misalignment between standard dimensionality reduction methods and downstream stochastic optimization by developing a prescriptive framework that minimizes suboptimality in optimization, showing it can be formulated as a distributionally-robust optimization problem and outperforming PCA in computational experiments on warehouse transshipment and vehicle repositioning problems.
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.