Minimizing Upper Confidence Bounds: A Data-Driven Framework for Stochastic Programming

TL;DR

This paper introduces a new data-driven framework for stochastic programming that minimizes an upper confidence bound on expected costs, enhancing robustness against epistemic uncertainty.

Contribution

It proposes the Average Percentile Upper Bound (APUB), a novel statistical construct with proven asymptotic correctness, and develops practical solution methods for APUB-based optimization.

Findings

Empirical results show APUB improves decision robustness.

APUB provides a statistically rigorous upper bound for means.

Framework enhances reliability and consistency in stochastic optimization.

Abstract

Stochastic programming is often challenged by epistemic uncertainty, where critical probability distributions are poorly characterized or unknown due to a lack of data. To address this, we pioneer a novel framework for stochastic programming that minimizes an upper confidence bound (UCB) on the expected random cost, acting as a robustness-seeking strategy. Our central contribution is the Average Percentile Upper Bound (APUB), a new statistical construct that serves as both a statistically rigorous upper bound for population means and an approximate risk metric for sample means. We rigorously prove the asymptotic correctness and consistency of APUB, establishing a reliable foundation for data-driven decision-making. We also develop practical solution methods, including a bootstrap sampling approximation method and an L-shaped method, to solve APUB optimization problems, with a specific focus on two-stage linear stochastic optimization with random recourse. Empirical demonstrations on a two-stage product mix problem reveal the significant benefits of our APUB optimization framework, which fortifies the process against epistemic uncertainty while reinforcing key decision-making attributes like reliability and consistency. The implementation and source code are available at https://github.com/8Wings/APUB-Optimization.