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Learning to Stop Cut Generation for Efficient Mixed-Integer Linear Programming

Haotian Ling, Zhihai Wang, Jie Wang

TL;DR

The paper tackles the efficiency bottleneck in MILP solving caused by when to stop cutting-plane generation. It recasts the stopping decision as a reinforcement-learning problem and introduces HYGRO, a hybrid graph representation model that merges dynamic MILP graph embeddings with static problem features. Trained with Evolutionary Strategies and integrated into SCIP, HYGRO learns effective stopping rules that adapt to instance characteristics, improving solving time and PDI by up to 31% on diverse benchmarks. This data-driven approach demonstrates strong generalization across problem sizes and problem types, offering a practical route to smarter, faster MILP solvers.

Abstract

Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), as they significantly tighten the dual bounds and improve the solving performance. A key problem for cuts is when to stop cuts generation, which is important for the efficiency of solving MILPs. However, many modern MILP solvers employ hard-coded heuristics to tackle this problem, which tends to neglect underlying patterns among MILPs from certain applications. To address this challenge, we formulate the cuts generation stopping problem as a reinforcement learning problem and propose a novel hybrid graph representation model (HYGRO) to learn effective stopping strategies. An appealing feature of HYGRO is that it can effectively capture both the dynamic and static features of MILPs, enabling dynamic decision-making for the stopping strategies. To the best of our knowledge, HYGRO is the first data-driven method to tackle the cuts generation stopping problem. By integrating our approach with modern solvers, experiments demonstrate that HYGRO significantly improves the efficiency of solving MILPs compared to competitive baselines, achieving up to 31% improvement.

Learning to Stop Cut Generation for Efficient Mixed-Integer Linear Programming

TL;DR

The paper tackles the efficiency bottleneck in MILP solving caused by when to stop cutting-plane generation. It recasts the stopping decision as a reinforcement-learning problem and introduces HYGRO, a hybrid graph representation model that merges dynamic MILP graph embeddings with static problem features. Trained with Evolutionary Strategies and integrated into SCIP, HYGRO learns effective stopping rules that adapt to instance characteristics, improving solving time and PDI by up to 31% on diverse benchmarks. This data-driven approach demonstrates strong generalization across problem sizes and problem types, offering a practical route to smarter, faster MILP solvers.

Abstract

Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), as they significantly tighten the dual bounds and improve the solving performance. A key problem for cuts is when to stop cuts generation, which is important for the efficiency of solving MILPs. However, many modern MILP solvers employ hard-coded heuristics to tackle this problem, which tends to neglect underlying patterns among MILPs from certain applications. To address this challenge, we formulate the cuts generation stopping problem as a reinforcement learning problem and propose a novel hybrid graph representation model (HYGRO) to learn effective stopping strategies. An appealing feature of HYGRO is that it can effectively capture both the dynamic and static features of MILPs, enabling dynamic decision-making for the stopping strategies. To the best of our knowledge, HYGRO is the first data-driven method to tackle the cuts generation stopping problem. By integrating our approach with modern solvers, experiments demonstrate that HYGRO significantly improves the efficiency of solving MILPs compared to competitive baselines, achieving up to 31% improvement.
Paper Structure (34 sections, 4 equations, 5 figures, 3 tables)

This paper contains 34 sections, 4 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Cutting Plane Method
  4. Stopping Strategies in Cutting Plane Method
  5. Fixed Cut Numbers (FCN)
  6. Fixed Cut Rounds (FCR)
  7. Stagnation Round Detection (SRD)
  8. Motivation Results
  9. Setup
  10. Results
  11. Method
  12. Motivation
  13. Problem Formulation
  14. State Space
  15. Action Space
  16. ...and 19 more sections

Figures (5)

  • Figure 1: The right gray dashed box illustrates main steps related to cutting planes that each node needs to undergo.
  • Figure 2: Lower values indicate better performance on the figures. Data for both figures was normalized. The legend in the right figure represents the instance numbers in the Anonymous dataset.
  • Figure 3: The red, blue, and yellow circles represent variable, constraint, and static features, respectively.
  • Figure 4: Diagram illustrating the structure of HYGRO. First, it employs a two-layer graph neural convolutional network to encode the dynamic part. Then, it aggregates the dynamic part with the static part and finally outputs the $ratio$. In summary, HYGRO takes the four features shown in the diagram as inputs and generates the $ratio$ value for the current node as output.
  • Figure 5: The y-axis shows HYGRO's solving time, and the x-axis indicates the best performance within the specified range of rounds (R). Points below the gray dashed line signify superior performance for HYGRO.