Learned Pairwise Deep Dual-Optimal Inequalities for Stabilizing Column Generation

2026-07-14

AI Summary

This work addresses the slow convergence of column generation in solving the capacitated vehicle routing problem with time windows, a challenge often caused by unstable dual solutions. To overcome this, the authors propose a novel framework that integrates machine learning with mathematical optimization. Specifically, they employ a classifier to predict pairwise ordering relationships among dual variables and encode these predictions as deep dual-optimal inequalities within the master problem. This approach is further enhanced by a graph-structured post-processing step and a selective slack restoration mechanism, which together preserve theoretical bounds while substantially accelerating computation. Experimental results demonstrate that the root-node solution time is reduced by 89.7% and 93.9% with average optimality gaps of only 1.3% and 0.5%, respectively. When the restoration mechanism is activated, speedups of 54.8% and 83.1% are achieved without any loss in solution quality.

Abstract

Column generation (CG) is central to many large-scale optimization algorithms, including branch-price-and-cut methods for vehicle routing problems, but unstable dual solutions can substantially slow its convergence. Existing deep dual-optimal inequalities can reduce this instability by restricting the dual space. Their construction, however, typically relies on problem-specific exchange arguments that are difficult to establish for routing problems with capacity limits, time windows, and other resource constraints. We introduce learned pairwise deep dual-optimal inequalities (L-PDDOIs), a learning framework that predicts pairwise orderings between dual variables and incorporates their primal counterparts directly into the master problem. To construct training labels, the framework samples optimal dual solutions and selects pairwise order relations that hold simultaneously on a sufficiently large common subset of the samples. A classifier then assigns a score to each candidate relation. Because conflicts and redundancies among the predicted relations can impair performance, graph-based postprocessing filters and compresses the candidate set before deployment. We further introduce a recovery procedure that selectively relaxes learned inequalities and provides a certificate when the baseline CG bound has been restored. On the main test sets for the capacitated vehicle routing problem and the vehicle routing problem with time windows, direct deployment of L-PDDOIs reduces the geometric mean root CG time by 89.7% and 93.9%, respectively, while incurring mean bound losses of only 1.3% and 0.5%. The recovery procedure retains corresponding time reductions of 54.8% and 83.1%, respectively, while guaranteeing no loss in the CG bound.

Problem

• column generation
• dual instability
• vehicle routing problem
• dual-optimal inequalities
• convergence acceleration

Innovation

• learned dual-optimal inequalities
• column generation stabilization
• pairwise ordering prediction
• graph-based postprocessing
• bound-preserving recovery