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A Quantile Neural Network Framework for Two-stage Stochastic Optimization

Antonio Alcántara, Carlos Ruiz, Calvin Tsay

TL;DR

This work proposes to model the distribution of the second-stage objective, specifically using a quantile neural network, and discusses optimization formulations for embedding the quantile neural network and demonstrates the effectiveness of the proposed framework using several computational case studies.

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.

A Quantile Neural Network Framework for Two-stage Stochastic Optimization

TL;DR

This work proposes to model the distribution of the second-stage objective, specifically using a quantile neural network, and discusses optimization formulations for embedding the quantile neural network and demonstrates the effectiveness of the proposed framework using several computational case studies.

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.
Paper Structure (25 sections, 15 equations, 3 figures, 8 tables, 2 algorithms)

This paper contains 25 sections, 15 equations, 3 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Two-stage Stochastic Optimization
  4. Quantile Neural Networks
  5. Mixed-Integer Formulations for ReLU Neural Networks
  6. Methodology
  7. Quantile Neural Network Methodology
  8. Data Generation
  9. Problem Formulation
  10. QNN surrogate model
  11. IQNN surrogate model
  12. Related Work
  13. Case Studies
  14. Experimental setup
  15. Two-stage stochastic optimization problems
  16. ...and 10 more sections

Figures (3)

  • Figure 1: QNN Structure for Two-stage Optimization Problems.
  • Figure 2: IQNN Output Layer Example.
  • Figure 3: Predictive (left) and prescriptive (right) performance of the (I)QNN framework with different numbers of training samples for the risk-averse CFLP-10-10 problem.