[Paper Review] A Machine Learning Approach to Two-Stage Adaptive Robust Optimization

Dimitris Bertsimas, Cheol Woo Kim|arXiv (Cornell University)|Jul 23, 2023

Risk and Portfolio Optimization4 citations

TL;DR

This paper proposes a machine learning approach to accelerate two-stage adaptive robust optimization (ARO) with binary here-and-now variables and polyhedral uncertainty sets. By pre-solving multiple ARO instances via column and constraint generation (CCG), the method extracts optimal strategies—comprising here-and-now decisions, worst-case scenarios, and wait-and-see decisions—and trains a machine learning model to predict these strategies, achieving speedups of several orders of magnitude over state-of-the-art algorithms while maintaining high accuracy.

ABSTRACT

We propose an approach based on machine learning to solve two-stage linear adaptive robust optimization (ARO) problems with binary here-and-now variables and polyhedral uncertainty sets. We encode the optimal here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the optimal wait-and-see decisions into what we denote as the strategy. We solve multiple similar ARO instances in advance using the column and constraint generation algorithm and extract the optimal strategies to generate a training set. We train a machine learning model that predicts high-quality strategies for the here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the wait-and-see decisions. We also introduce an algorithm to reduce the number of different target classes the machine learning algorithm needs to be trained on. We apply the proposed approach to the facility location, the multi-item inventory control and the unit commitment problems. Our approach solves ARO problems drastically faster than the state-of-the-art algorithms with high accuracy.

Motivation & Objective

• To address the high computational cost of solving two-stage adaptive robust optimization (ARO) problems in real-time settings.
• To develop a machine learning framework that predicts optimal strategies—including here-and-now decisions, worst-case scenarios, and wait-and-see decisions—for ARO problems.
• To reduce the number of distinct target classes in the machine learning training process, which is critical due to the combinatorial explosion of uncertainty set realizations.
• To enable fast, near-optimal solutions for ARO problems in applications like facility location, inventory control, and unit commitment, where real-time decision-making is essential.

Proposed method

• The method uses the column and constraint generation (CCG) algorithm to pre-solve multiple similar ARO instances and extract their optimal strategies, forming a supervised training dataset.
• A machine learning model is trained to predict three components: the optimal here-and-now decisions, the worst-case scenarios corresponding to those decisions, and the optimal wait-and-see decisions.
• A novel class reduction technique is introduced to limit the number of distinct target classes, improving generalization and reducing model complexity in high-dimensional uncertainty settings.
• The approach encodes the optimal strategy as a joint prediction of binary here-and-now variables, worst-case uncertainty realizations, and continuous wait-and-see variables.
• The model is trained on a dataset of pre-solved ARO instances, enabling fast inference during online deployment.
• The method is evaluated on three real-world problems: facility location, multi-item inventory control, and unit commitment, using benchmark formulations.

Experimental results

Research questions

• RQ1 Can machine learning be effectively used to predict the full optimal strategy—here-and-now decisions, worst-case scenarios, and wait-and-see decisions—in two-stage adaptive robust optimization?
• RQ2 How can the number of distinct target classes in the machine learning prediction task be reduced to improve model scalability and performance in high-dimensional uncertainty sets?
• RQ3 To what extent can a learned strategy model achieve speedups over traditional exact algorithms like CCG while maintaining solution quality in ARO problems?
• RQ4 Can the proposed approach be generalized across different ARO applications, such as facility location, inventory control, and unit commitment, with consistent performance gains?

Key findings

• The proposed machine learning approach solves two-stage ARO problems in milliseconds, achieving speedups of several orders of magnitude compared to the state-of-the-art CCG algorithm.
• The method maintains high solution accuracy, with predicted strategies closely matching the optimal solutions obtained via CCG on all tested problem instances.
• The class reduction technique significantly improves model generalization and reduces training complexity, especially in problems with large uncertainty sets.
• The approach outperforms existing machine learning methods for mixed-integer convex optimization by extending the framework to infinite-dimensional ARO problems with functional wait-and-see decisions.
• Empirical results on facility location, multi-item inventory control, and unit commitment problems confirm the method’s robustness and scalability across diverse applications.
• The model generalizes well to unseen instances, demonstrating strong generalization performance even when tested on ARO instances with uncertainty sets not present in the training data.