LEARNING CUT SELECTION FOR MIXED-INTEGER LINEAR PROGRAMMING VIA HIERARCHICAL SEQUENCE MODEL

Abstract

Cutting planes (cuts) are important for solving mixed-integer linear programs (MILPs), which formulate a wide range of important real-world applications. Cut selection-which aims to select a proper subset of the candidate cuts to improve the efficiency of solving MILPs-heavily depends on (P1) which cuts should be preferred, and (P2) how many cuts should be selected. Although many modern MILP solvers tackle (P1)-(P2) by manually designed heuristics, machine learning offers a promising approach to learn more effective heuristics from MILPs collected from specific applications. However, many existing learning-based methods focus on learning which cuts should be preferred, neglecting the importance of learning the number of cuts that should be selected. Moreover, we observe from extensive empirical results that (P3) what order of selected cuts should be preferred has a significant impact on the efficiency of solving MILPs as well. To address this challenge, we propose a novel hierarchical sequence model (HEM) to learn cut selection policies via reinforcement learning. Specifically, HEM consists of a two-level model: (1) a higher-level model to learn the number of cuts that should be selected, (2) and a lower-level model-that formulates the cut selection task as a sequence to sequence learning problem-to learn policies selecting an ordered subset with the size determined by the higher-level model. To the best of our knowledge, HEM is the first method that can tackle (P1)-(P3) in cut selection simultaneously from a data-driven perspective. Experiments show that HEM significantly improves the efficiency of solving MILPs compared to human-designed and learning-based baselines on both synthetic and large-scale real-world MILPs, including MI-PLIB 2017. Moreover, experiments demonstrate that HEM well generalizes to MILPs that are significantly larger than those seen during training.

1. INTRODUCTION

Mixed-integer linear programming (MILP) is a general optimization formulation for a wide range of important real-world applications, such as supply chain management (Paschos, 2014), production planning (Jünger et al., 2009) , scheduling (Chen, 2010), facility location (Farahani & Hekmatfar, 2009) , bin packing (Nair et al., 2020) , etc. A standard MILP takes the form of z * ≜ min x {c ⊤ x|Ax ≤ b, x ∈ R n , x j ∈ Z for all j ∈ I}, where c ∈ R n , A ∈ R m×n , b ∈ R m , x j denotes the j-th entry of vector x, I ⊆ {1, . . . , n} denotes the set of indices of integer variables, and z * denotes the optimal objective value of the problem in (1). However, MILPs can be extremely hard to solve as they are N P-hard problems (Bixby et al., 2004) . To solve MILPs, many modern MILP solvers (Gurobi, 2021; Bestuzheva et al., 2021; FICO Xpress, 2020) employ a branch-and-bound tree search algorithm (Land & Doig, 2010) , in which a linear programming (LP) relaxation of a MILP (the problem in (1) or its subproblems) is solved at each node. To further enhance the performance of the tree search algorithm, cutting planes (cuts) (Gomory, 1960) are introduced to tighten the LP relaxations (Achterberg, 2007; Bengio et al., 2021) . Existing work on cuts falls into two categories: cut generation and cut selection (Turner et al., 2022) . Cut generation aims to generate cuts, i.e., valid linear inequalities that tighten the LP relaxations (Achterberg, 2007) . However, adding all the generated cuts to the LP relaxations can pose a computational problem (Wesselmann & Stuhl, 2012) . To further improve the efficiency of solving MILPs, cut selection is proposed to select a proper subset of the generated cuts (Wesselmann & Stuhl, 2012) . In this paper, we focus on the cut selection problem, which has a significant impact on the overall solver performance (Achterberg, 2007; Tang et al., 2020; Paulus et al., 2022) . Cut selection heavily depends on (P1) which cuts should be preferred, and (P2) how many cuts should be selected (Achterberg, 2007; Dey & Molinaro, 2018b ). Moreover, we observe from extensive empirical results that (P3) what order of selected cuts should be preferred significantly impacts the efficiency of solving MILPs as well (see Section 3). Second, they do not take into account the interaction among cuts when learning which cuts should be preferred, as they score each cut independently. As a result, they struggle to select cuts that complement each other nicely, which could severely hinder the efficiency of solving MILPs (Dey & Molinaro, 2018b) . Indeed, we empirically show that they tend to select many similar cuts with high scores (see Experiment 4 in Section 5). To address the aforementioned challenges, we propose a novel hierarchical sequence model (HEM) to learn cut selection policies via reinforcement learning. To the best of our knowledge, HEM is the first learning-based method that can tackle (P1)-(P3) simultaneously by proposing a two-level model. Specifically, HEM is comprised of (1) a higher-level model to learn the number of cuts that should be selected, (2) and a lower-level model to learn policies selecting an ordered subset with the size determined by the higher-level model. The lower-level model formulates the cut selection task as a sequence to sequence learning problem, leading to two major advantages. First, the sequence model is popular in capturing the underlying order information (Vinyals et al., 2016) , which is critical for tackling (P3). Second, the sequence model can well capture the interaction among cuts, as it models the joint conditional probability of the selected cuts given an input sequence of the candidate cuts. As a result, experiments show that HEM significantly outperforms human-designed and learning-based baselines in terms of solving efficiency on three synthetic MILP problems and seven challenging MILP problems. The challenging MILP problems include some benchmarks from MIPLIB 2017 (Gleixner et al., 2021) and large-scale real-world production planning problems. Our results demonstrate the strong ability to enhance modern MILP solvers with our proposed HEM in real-world applications. Moreover, experiments demonstrate that HEM can well generalize to MILPs that are significantly larger than those seen during training.

2. BACKGROUND

Cutting planes. Given the MILP problem in (1), we drop all its integer constraints to obtain its linear programming (LP) relaxation, which takes the form of z * LP ≜ min x {c ⊤ x|Ax ≤ b, x ∈ R n }. (2)



). Many modern MILP solvers(Gurobi, 2021; Bestuzheva et al., 2021; FICO Xpress, 2020)  tackle (P1)-(P2) by hard-coded heuristics designed by experts. However, hard-coded heuristics do not take into account underlying patterns among MILPs collected from certain types of real-world applications, e.g., day-to-day production planning, bin packing, and vehicle routing problems(Pochet & Wolsey, 2006; Laporte, 2009;  Nair et al., 2020). To further improve the efficiency of MILP solvers, recent methods(Tang et al.,  2020; Paulus et al., 2022; Huang et al., 2022)  propose to learn cut selection policies via machine learning, especially reinforcement learning. They offer promising approaches to learn more effec-

funding

/mindspore/models/tree/ master/research/l2o/hem-learning-to-cut (MindSpore version). * Equal contribution. This work was done when Zhihai Wang was an intern at Huawei Noah's Ark Lab.

