Predictive Dual Smoothing for Column Generation

2026-09-28

AI Summary

This study addresses the slow convergence of column generation algorithms in large-scale linear programming caused by dual solution oscillation. To mitigate this issue, we propose a predictive dual smoothing method that introduces supervised learning into dual stabilization for the first time. Specifically, an offline-trained model predicts future dual values to guide the pricing subproblem, replacing conventional strategies that rely solely on historical information, while preserving exact verification mechanisms to guarantee solution correctness. Experimental results demonstrate that the proposed approach significantly reduces both the number of generated columns and computational time on mixed-integer programming tasks such as the cutting stock problem. Furthermore, it exhibits strong generalization capabilities on out-of-distribution instances, achieving overall performance superior to existing state-of-the-art methods.

Abstract

Solving large-scale linear programs efficiently is an important challenge in many optimization settings. A key technique is column generation, which alternates between solving the master problem over a restricted subset of the variables, and using a pricing subproblem to identify new variables to add. The pricing subproblem is guided by the dual solution of the current restricted master problem, but oscillations in these dual solutions can substantially slow convergence. Dual stabilization methods address this issue. Dual smoothing is a common stabilization method, which guides the pricing subproblem using a combination of the current dual solution and duals from previous iterations. However, while past dual solutions can stabilize the dual trajectory, they do not necessarily guide pricing towards useful new variables. We therefore introduce predictive dual smoothing, which instead combines the current dual solution with a learned prediction of future duals to steer pricing towards variables that are more useful in subsequent iterations. The predictor is trained offline using supervision extracted from standard column generation trajectories and is used only to modify the pricing subproblem's objective function, while exact reduced-cost checks and fallback pricing with the unsmoothed duals preserve correctness. Experiments on cutting stock and generalized assignment problems show that predictive dual smoothing substantially reduces generated columns and wall-clock time relative to standard column generation and existing classical and learned stabilization methods. These gains extend to out-of-distribution instance sizes, and predictive smoothing provides further improvements when combined with strong classical stabilization.

Problem

• Column Generation
• Large-scale Linear Programming
• Dual Smoothing
• Dual Stabilization
• Pricing Subproblem

Innovation

• Column Generation
• Predictive Dual Smoothing
• Dual Stabilization
• Machine Learning for Optimization
• Pricing Subproblem