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CLCR: Contrastive Learning-based Constraint Reordering for Efficient MILP Solving

Shuli Zeng, Mengjie Zhou, Sijia Zhang, Yixiang Hu, Feng Wu, Xiang-Yang Li

TL;DR

This work tackles the sensitivity of MILP solving to constraint ordering byIntroducing CLCR, which clusters constraints by structure, reorders them with a pointer-network, and trains the orderings via contrastive learning to minimize solving time while preserving problem equivalence. The method leverages a clustering step, a sequence model for cluster ordering, and a contrastive objective to prefer permutations that shorten $t_{\text{solve}}$, avoiding unstable RL training. Empirical results on nine datasets, including SurrogateLIB and traditional MILPs, show substantial gains in total time and LP iterations across baselines, with notable reductions in presolve time and node exploration. The approach suggests a practical, generalizable path to improve MILP performance through learned, data-driven constraint reformulation, while recognizing symmetry-related limitations and proposing a discriminator to gate reordering decisions.

Abstract

Constraint ordering plays a critical role in the efficiency of Mixed-Integer Linear Programming (MILP) solvers, particularly for large-scale problems where poorly ordered constraints trigger increased LP iterations and suboptimal search trajectories. This paper introduces CLCR (Contrastive Learning-based Constraint Reordering), a novel framework that systematically optimizes constraint ordering to accelerate MILP solving. CLCR first clusters constraints based on their structural patterns and then employs contrastive learning with a pointer network to optimize their sequence, preserving problem equivalence while improving solver efficiency. Experiments on benchmarks show CLCR reduces solving time by 30% and LP iterations by 25% on average, without sacrificing solution accuracy. This work demonstrates the potential of data-driven constraint ordering to enhance optimization models, offering a new paradigm for bridging mathematical programming with machine learning.

CLCR: Contrastive Learning-based Constraint Reordering for Efficient MILP Solving

TL;DR

This work tackles the sensitivity of MILP solving to constraint ordering byIntroducing CLCR, which clusters constraints by structure, reorders them with a pointer-network, and trains the orderings via contrastive learning to minimize solving time while preserving problem equivalence. The method leverages a clustering step, a sequence model for cluster ordering, and a contrastive objective to prefer permutations that shorten , avoiding unstable RL training. Empirical results on nine datasets, including SurrogateLIB and traditional MILPs, show substantial gains in total time and LP iterations across baselines, with notable reductions in presolve time and node exploration. The approach suggests a practical, generalizable path to improve MILP performance through learned, data-driven constraint reformulation, while recognizing symmetry-related limitations and proposing a discriminator to gate reordering decisions.

Abstract

Constraint ordering plays a critical role in the efficiency of Mixed-Integer Linear Programming (MILP) solvers, particularly for large-scale problems where poorly ordered constraints trigger increased LP iterations and suboptimal search trajectories. This paper introduces CLCR (Contrastive Learning-based Constraint Reordering), a novel framework that systematically optimizes constraint ordering to accelerate MILP solving. CLCR first clusters constraints based on their structural patterns and then employs contrastive learning with a pointer network to optimize their sequence, preserving problem equivalence while improving solver efficiency. Experiments on benchmarks show CLCR reduces solving time by 30% and LP iterations by 25% on average, without sacrificing solution accuracy. This work demonstrates the potential of data-driven constraint ordering to enhance optimization models, offering a new paradigm for bridging mathematical programming with machine learning.

Paper Structure

This paper contains 21 sections, 8 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed Integer Linear Programming
  4. Core Mechanisms of MILP Solvers
  5. Related Work
  6. Reformulating Linear Programming
  7. Learning Order in Branch-and-Bound
  8. Motivation
  9. Order Matters
  10. Motivation Results
  11. Proposed Method
  12. Clustering and Representing Constraints
  13. Pointer Network-Based Reordering
  14. Contrastive Learning for Constraint Reordering
  15. Experiments
  16. ...and 6 more sections

Figures (2)

  • Figure 1: We evaluate the impact of random constraint reordering on solver performance using five different cases. Each bar represents the solver time across 100 random seeds, with the standard deviation and percentage of variation indicated.
  • Figure 2: Illustration of the constraint representation, clustering, and reordering process. Constraints are represented as feature vectors and grouped into clusters using K-means. A pointer network is then used to determine the optimal ordering of the clusters, which is applied to reorder the constraints in the original MILP problem.