发表机构
KU Leuven; TU Delft; Cornell Tech（鲁汶大学; 代尔夫特理工大学; 康奈尔科技校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对列生成中对偶振荡导致的收敛慢问题，提出预测性对偶平滑，用学习预测的未来对偶引导定价，实验表明大幅减少列数和时间。
AI 中文摘要
在许多优化场景中，高效求解大规模线性规划是一个重要挑战。一种关键技术是列生成，它交替进行以下操作：在变量的受限子集上求解主问题，并使用定价子问题来识别要添加的新变量。定价子问题由当前受限主问题的对偶解引导，但这些对偶解的振荡会显著减慢收敛速度。对偶稳定化方法解决了这一问题。对偶平滑是一种常见的稳定化方法，它使用当前对偶解与先前迭代的对偶解的组合来引导定价子问题。然而，虽然过去的对偶解可以稳定对偶轨迹，但它们不一定引导定价走向有用的新变量。因此，我们引入了预测性对偶平滑，它转而将当前对偶解与未来对偶的学习预测相结合，以引导定价走向在后续迭代中更有用的变量。预测器使用从标准列生成轨迹中提取的监督信号进行离线训练，并且仅用于修改定价子问题的目标函数，而精确的降价检查和带有未平滑对偶的回退定价则保证了正确性。在切割下料和广义分配问题上的实验表明，相对于标准列生成以及现有的经典和学习型稳定化方法，预测性对偶平滑大幅减少了生成的列数和墙钟时间。这些收益扩展到分布外实例规模，并且当与强经典稳定化结合时，预测性平滑提供了进一步的改进。
英文摘要
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.
