A Quantile Neural Network Framework for Two-stage Stochastic Optimization

TL;DR

This paper introduces a novel quantile neural network framework for two-stage stochastic optimization, allowing for distributional modeling of second-stage costs and enabling risk-aware decision-making beyond expected value.

Contribution

It proposes embedding a quantile neural network to approximate the distribution of second-stage costs, enhancing uncertainty modeling in stochastic optimization.

Findings

Effective in capturing the distribution of second-stage costs.

Enables optimization of risk measures like CVaR.

Demonstrated success on mixed-integer problems.

Abstract

Two-stage stochastic programming is a popular framework for optimization under uncertainty, where decision variables are split between first-stage decisions, and second-stage (or recourse) decisions, with the latter being adjusted after uncertainty is realized. These problems are often formulated using Sample Average Approximation (SAA), where uncertainty is modeled as a finite set of scenarios, resulting in a large "monolithic" problem, i.e., where the model is repeated for each scenario. The resulting models can be challenging to solve, and several problem-specific decomposition approaches have been proposed. An alternative approach is to approximate the expected second-stage objective value using a surrogate model, which can then be embedded in the first-stage problem to produce good heuristic solutions. In this work, we propose to instead model the distribution of the second-stage objective, specifically using a quantile neural network. Embedding this distributional approximation enables capturing uncertainty and is not limited to expected-value optimization, e.g., the proposed approach enables optimization of the Conditional Value at Risk (CVaR). We discuss optimization formulations for embedding the quantile neural network and demonstrate the effectiveness of the proposed framework using several computational case studies including a set of mixed-integer optimization problems.