Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning to Cut via Hierarchical Sequence/Set Model for Efficient Mixed-Integer Programming

Jie Wang, Zhihai Wang, Xijun Li, Yufei Kuang, Zhihao Shi, Fangzhou Zhu, Mingxuan Yuan, Jia Zeng, Yongdong Zhang, Feng Wu

TL;DR

The paper tackles the MILP cut-selection problem by addressing what cuts to pick, how many to pick, and in what order. It introduces HEM, a bi-level hierarchical model that first predicts the number of cuts and then selects an ordered subset via Seq2Seq/Set2Seq policies, enabling explicit modeling of order and interaction among cuts. Empirical results across nine benchmarks, including large-scale Huawei and Google problems, show that HEM and its extension HEM++ substantially improve solving time and the primal-dual gap integral compared to strong baselines and prior learning methods. The authors also extract human-readable order rules from learned policies and demonstrate applicability to real-world solvers, indicating practical impact for data-driven MILP solving.

Abstract

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.

Learning to Cut via Hierarchical Sequence/Set Model for Efficient Mixed-Integer Programming

TL;DR

The paper tackles the MILP cut-selection problem by addressing what cuts to pick, how many to pick, and in what order. It introduces HEM, a bi-level hierarchical model that first predicts the number of cuts and then selects an ordered subset via Seq2Seq/Set2Seq policies, enabling explicit modeling of order and interaction among cuts. Empirical results across nine benchmarks, including large-scale Huawei and Google problems, show that HEM and its extension HEM++ substantially improve solving time and the primal-dual gap integral compared to strong baselines and prior learning methods. The authors also extract human-readable order rules from learned policies and demonstrate applicability to real-world solvers, indicating practical impact for data-driven MILP solving.

Abstract

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.
Paper Structure (87 sections, 6 theorems, 25 equations, 6 figures, 35 tables, 2 algorithms)

This paper contains 87 sections, 6 theorems, 25 equations, 6 figures, 35 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Cutting planes
  5. Branch-and-cut
  6. Primal-dual gap integral
  7. Motivating results
  8. Order matters
  9. Ratio matters
  10. Learning Cut Selection via Hierarchical Sequence Model (HEM)
  11. Reinforcement Learning Formulation
  12. Hierarchical Sequence Model
  13. Motivation
  14. Policy network architecture
  15. Instantiation of the policy network
  16. ...and 72 more sections

Key Result

Proposition 1

Given the cut selection policy $\pi_{\theta}(a_k|s) = \mathbb{E}_{k\sim \pi^h_{\theta_1}(\cdot|s)}[\pi^l_{\theta_2}(a_k|s,k)]$ and the training objective (eq:obj), the hierarchical policy gradient takes the form of

Figures (6)

  • Figure 1: (a)-(b) We design two cut selection heuristics, namely RandomAll and RandomNV (see Section \ref{['sec:order']} for details), which both add the same subset of cuts in random order for a given MILP. The results in (a) and (b) show that adding the same selected cuts in different order leads to variable overall solver performance. (c)-(d) We use the Normalized Violation (NV) heuristics cut_ranking in the following experiments. The results in (c) show that the performance of NV varies widely with the given ratios across different datasets. The results in (d) show that the performance of NV varies widely with the given ratios across different instances from the Anonymous dataset.
  • Figure 2: Illustration of our proposed RL framework for learning cut selection policies. We formulate a MILP solver as the environment and the HEM as the agent. Moreover, we train HEM via hierarchical policy gradients.
  • Figure 3: The instantiation of HEM.
  • Figure 4: We instantiate the Set2Seq model via a multi-head attention encoder and a pointer decoder.
  • Figure 5: We perform principal component analysis on the cuts selected by HEM-ratio and SBP. Colored points illustrate the reduced cut features. The area covered by the dashed lines represents the diversity of selected cuts. The results show that HEM-ratio selects much more diverse cuts than SBP.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Proposition 1
  • Proposition 2
  • Definition 1
  • Proposition 3
  • Theorem 1
  • proof
  • proof
  • proof
  • Definition 2
  • Lemma 1
  • ...and 4 more