用于稳定列生成的学习成对深度对偶最优不等式
Learned Pairwise Deep Dual-Optimal Inequalities for Stabilizing Column Generation
- Department of Civil Engineering, Tsinghua University, Beijing 100084, China（清华大学土木工程系）
- MIT Sloan School of Management, Massachusetts Institute of Technology, 100 Main Street, Cambridge, MA 02142, USA（麻省理工学院斯隆管理学院）
- Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA（佛罗里达大学工业与系统工程系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究针对列生成中对偶解不稳定问题，提出学习成对深度对偶最优不等式框架，通过预测对偶变量排序、构建训练标签、后处理及恢复过程，在车辆路由问题测试集上大幅减少CG时间且保证边界无损失。
AI中文摘要：
列生成（CG）是许多大规模优化算法的核心，但不稳定的对偶解会显著减缓其收敛速度。现有深度对偶最优不等式可通过限制对偶空间来减少这种不稳定性，但其构建通常依赖于特定问题的交换论证，对于有容量限制、时间窗口等资源约束的路由问题难以建立。我们引入学习成对深度对偶最优不等式（L-PDDOIs），它预测对偶变量之间的成对排序并将其原始对应项直接纳入主问题。通过采样最优对偶解构建训练标签，用分类器为候选关系打分，利用基于图的后处理过滤和压缩候选集。还引入恢复过程，选择性地放宽学习到的不等式并在恢复基线CG边界时提供证书。在有容量车辆路由问题和带时间窗口车辆路由问题的主要测试集上，直接部署L-PDDOIs分别将几何平均根CG时间减少89.7%和93.9%，平均边界损失仅为1.3%和0.5%。恢复过程分别保持相应的时间减少54.8%和83.1%，同时保证CG边界无损失。
英文摘要：
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.
