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

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.

