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Contextual Scenario Generation for Two-Stage Stochastic Programming

David Islip, Roy H. Kwon, Sanghyeon Bae, Woo Chang Kim

TL;DR

This work tackles decision-making under uncertainty in contextual two-stage stochastic programs by learning a mapping from context to a small surrogate scenario set. It introduces distributional contextual scenario generation (DCSG) that minimizes a conditional distribution distance via MMD, and problem-driven contextual scenario generation (PCSG) that optimizes downstream objective quality through a learned Loss-Net approximator of the task loss ell_opt. The framework is validated on Newsvendor, CEP1, Portfolio CVaR, and Multidimensional Newsvendor with Substitution, showing that appropriately regularized, problem-driven mappings yield high-quality decisions with notable runtime savings, especially when a larger surrogate set is used. The results demonstrate that combining distributional alignment with downstream-task learning enables robust, context-aware scenario generation that scales to diverse problem structures while providing practical speedups for repeated contextual decision problems.

Abstract

Two-stage stochastic programs (2SPs) are important tools for making decisions under uncertainty. Decision-makers use contextual information to generate a set of scenarios to represent the true conditional distribution. However, the number of scenarios required is a barrier to implementing 2SPs, motivating the problem of generating a small set of surrogate scenarios that yield high-quality decisions when they represent uncertainty. Current scenario generation approaches do not leverage contextual information or do not address computational concerns. In response, we propose contextual scenario generation (CSG) to learn a mapping between the context and a set of surrogate scenarios of user-specified size. First, we propose a distributional approach that learns the mapping by minimizing a distributional distance between the predicted surrogate scenarios and the true contextual distribution. Second, we propose a task-based approach that aims to produce surrogate scenarios that yield high-quality decisions. The task-based approach uses neural architectures to approximate the downstream objective and leverages the approximation to search for the mapping. The proposed approaches apply to various problem structures and loosely only require efficient solving of the associated subproblems and 2SPs defined on the reduced scenario sets. Numerical experiments demonstrating the effectiveness of the proposed methods are presented.

Contextual Scenario Generation for Two-Stage Stochastic Programming

TL;DR

This work tackles decision-making under uncertainty in contextual two-stage stochastic programs by learning a mapping from context to a small surrogate scenario set. It introduces distributional contextual scenario generation (DCSG) that minimizes a conditional distribution distance via MMD, and problem-driven contextual scenario generation (PCSG) that optimizes downstream objective quality through a learned Loss-Net approximator of the task loss ell_opt. The framework is validated on Newsvendor, CEP1, Portfolio CVaR, and Multidimensional Newsvendor with Substitution, showing that appropriately regularized, problem-driven mappings yield high-quality decisions with notable runtime savings, especially when a larger surrogate set is used. The results demonstrate that combining distributional alignment with downstream-task learning enables robust, context-aware scenario generation that scales to diverse problem structures while providing practical speedups for repeated contextual decision problems.

Abstract

Two-stage stochastic programs (2SPs) are important tools for making decisions under uncertainty. Decision-makers use contextual information to generate a set of scenarios to represent the true conditional distribution. However, the number of scenarios required is a barrier to implementing 2SPs, motivating the problem of generating a small set of surrogate scenarios that yield high-quality decisions when they represent uncertainty. Current scenario generation approaches do not leverage contextual information or do not address computational concerns. In response, we propose contextual scenario generation (CSG) to learn a mapping between the context and a set of surrogate scenarios of user-specified size. First, we propose a distributional approach that learns the mapping by minimizing a distributional distance between the predicted surrogate scenarios and the true contextual distribution. Second, we propose a task-based approach that aims to produce surrogate scenarios that yield high-quality decisions. The task-based approach uses neural architectures to approximate the downstream objective and leverages the approximation to search for the mapping. The proposed approaches apply to various problem structures and loosely only require efficient solving of the associated subproblems and 2SPs defined on the reduced scenario sets. Numerical experiments demonstrating the effectiveness of the proposed methods are presented.

Paper Structure

This paper contains 35 sections, 3 theorems, 49 equations, 10 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contextual Optimization for Stochastic Programming
  4. Scenario Generation
  5. Learning-based Approaches for Stochastic Programs
  6. Proposed Methodology
  7. Preliminaries: Distributional Scenario Generation
  8. Distributional Contextual Scenario Generation
  9. Preliminaries: Problem-driven Scenario Generation
  10. Problem-driven Contextual Scenario Generation
  11. Solution Methodology
  12. Experiments
  13. Newsvendor
  14. Capacity Planning (CEP1)
  15. Portfolio Optimization
  16. ...and 20 more sections

Key Result

Theorem 1

If (i) $k(\cdot,\cdot )$ is characteristic and measurable, (ii) $\mathbb{E}_{\omega \sim \mathbb{P}_{\omega|\boldsymbol{x}}}[k(\omega, \omega)] < \infty$ and (iii) $\mathbb{E}_{\omega \sim \mathbb{P}_{\boldsymbol{f}(\boldsymbol{x})}}[k(\omega, \omega)] < \infty$ for all $\boldsymbol{x} \in \mathcal{

Figures (10)

  • Figure 1: Optimization with Scenario Generation
  • Figure 2: Contextual Scenario Generation and Optimization
  • Figure 3: Loss-Net Architecture
  • Figure 4: Replacing $\ell_{\text{opt}}$ with $E_{\psi}$
  • Figure 5: Visual representation of the DCSG and PCSG training procedures
  • ...and 5 more figures

Theorems & Definitions (4)

  • Theorem 1: Theorem 4, from huang2022evaluating
  • Proposition 1
  • Proposition 2: Proposition 1 from (Tabaghi and Wang 2024)
  • proof