Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), which formulate many important real-world applications. Cut selection heavily depends on (P1) which cuts to prefer and (P2) how many cuts to select. Although modern MILP solvers tackle (P1)-(P2) by human-designed heuristics, machine learning carries the potential to learn more effective heuristics. However, many existing learning-based methods learn which cuts to prefer, neglecting the importance of learning how many cuts to select. Moreover, we observe that (P3) what order of selected cuts to prefer significantly impacts the efficiency of MILP solvers as well. To address these challenges, we propose a novel hierarchical sequence/set model (HEM) to learn cut selection policies. Specifically, HEM is a bi-level model: (1) a higher-level module that learns how many cuts to select, (2) and a lower-level module -- that formulates the cut selection as a sequence/set to sequence learning problem -- to learn policies selecting an ordered subset with the cardinality determined by the higher-level module. To the best of our knowledge, HEM is the first data-driven methodology that well tackles (P1)-(P3) simultaneously. Experiments demonstrate that HEM significantly improves the efficiency of solving MILPs on eleven challenging MILP benchmarks, including two Huawei's real problems.

翻译：切割平面（割）在求解混合整数线性规划（MILP）中发挥着关键作用，而MILP建模了许多重要的现实应用。割的选择严重依赖于（P1）优先选择哪些割以及（P2）选择多少割。尽管现代MILP求解器通过人工设计的启发式方法处理（P1）-（P2）问题，但机器学习具有学习更有效启发式方法的潜力。然而，许多现有的基于学习的方法仅关注学习优先选择哪些割，忽视了学习选择多少割的重要性。此外，我们观察到（P3）所选割的排列顺序同样显著影响MILP求解器的效率。为解决这些挑战，我们提出了一种新颖的分层序列/集合模型（HEM）来学习割选择策略。具体而言，HEM是一个双层模型：（1）高层模块学习选择多少割，（2）低层模块将割选择表述为序列/集合到序列的学习问题，从而学习选择有序子集的策略，其基数由高层模块确定。据我们所知，HEM是首个能够同时妥善处理（P1）-（P3）问题的数据驱动方法。实验表明，HEM在十一个具有挑战性的MILP基准测试（包括两个华为实际案例）中显著提升了MILP求解效率。