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

TL;DR

This paper introduces a novel data-driven approach using piecewise affine decision rules within an ERM framework for stochastic programming with covariate data, providing theoretical guarantees and demonstrating superior empirical performance.

Contribution

It develops a nonconvex PADR-based ERM method with proven consistency, an advanced algorithm for solving it, and shows its effectiveness across various stochastic programming problems.

Findings

Proven nonasymptotic and asymptotic consistency of the PADR-ERM model.

The algorithm converges to stationary points with complexity guarantees.

Numerical results show lower costs and faster computation compared to existing methods.

Abstract

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 and asymptotic consistency result for constrained ones. To solve the nonconvex and nondifferentiable ERM problem, we develop an enhanced stochastic majorization-minimization algorithm and establish the asymptotic convergence to (composite strong) directional stationarity along with complexity 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. Our numerical study demonstrates the superior performance of PADR-based ERM methods compared to state-of-the-art approaches under various settings, with significantly lower costs, less computation time, and robustness to feature dimensions and nonlinearity of the underlying dependency.