Column generation (CG) is one of the most successful approaches for solving large-scale linear programming (LP) problems. Given an LP with a prohibitively large number of variables (i.e., columns), the idea of CG is to explicitly consider only a subset of columns and iteratively add potential columns to improve the objective value. While adding the column with the most negative reduced cost can guarantee the convergence of CG, it has been shown that adding multiple columns per iteration rather than a single column can lead to faster convergence. However, it remains a challenge to design a multiple-column selection strategy to select the most promising columns from a large number of candidate columns. In this paper, we propose a novel reinforcement-learning-based (RL) multiple-column selection strategy. To the best of our knowledge, it is the first RL-based multiple-column selection strategy for CG. The effectiveness of our approach is evaluated on two sets of problems: the cutting stock problem and the graph coloring problem. Compared to several widely used single-column and multiple-column selection strategies, our RL-based multiple-column selection strategy leads to faster convergence and achieves remarkable reductions in the number of CG iterations and runtime.

翻译：列生成（CG）是解决大规模线性规划（LP）问题最成功的方法之一。针对变量（即列）数量过大的线性规划问题，CG的基本思想是仅显式考虑一个列子集，并通过迭代添加潜在列来改进目标值。尽管添加具有最负简约成本的列可以保证CG的收敛性，但研究表明，每轮迭代添加多列而非单列能够加速收敛。然而，如何设计一种多列选择策略从大量候选列中筛选出最有潜力的列仍是一项挑战。本文提出了一种基于强化学习（RL）的新型多列选择策略。据我们所知，这是首个针对CG的RL多列选择策略。我们通过两类问题（下料问题和图着色问题）评估了该方法的有效性。与几种广泛使用的单列和多列选择策略相比，我们的RL多列选择策略实现了更快的收敛速度，并在CG迭代次数和运行时间方面取得了显著减少。