Data-Driven Piecewise Affine Decision Rules for Stochastic Programming with Covariate Information

Focusing on stochastic programming (SP) with covariate information, this paper proposes an empirical risk minimization (ERM) method embedded within a nonconvex piecewise affine decision rule (PADR), which aims to learn the direct mapping from features to optimal decisions. We establish the nonasymptotic consistency result of our PADR-based ERM model for unconstrained problems, which illustrates the role of piece number in balancing the trade-off between the approximation and estimation errors. To solve the nonconvex and nondifferentiable ERM problem, we develop an enhanced stochastic majorization-minimization algorithm and establish the convergence to (composite strong) directional stationarity, along with convergence rate analysis. We show that the proposed PADR-based ERM method applies to a broad class of nonconvex SP problems with theoretical consistency guarantees and computational tractability. Numerical experiments show that PADR significantly lowers costs with less computation time, and is more robust to feature dimensions and nonlinearity of the underlying dependency.

Funding: J. Liu is supported by the National Natural Science Foundation of China [Grants 72201151 and 72188101]. X. Zhao is supported by the National Natural Science Foundation of China [Grant 72271136].

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2023.0175.