Machine learning improves solving mixed integer linear programs faster
Reformulation-Contrastive Learning for Mixed Integer Programs
Machine Learning
Summary
Many real-world problems can be solved using mixed-integer linear programs (MILPs), but they can be slow for computers to solve. The authors found a way to teach computers to understand different ways these problems are written without needing help from solvers. They trained a model to recognize how changes in the problem's structure affect its parts, so it learns useful patterns automatically. This helps computers predict important properties of MILPs more efficiently, leading to faster and smarter solving.
What this means in practice
- •For optimization engineers: Improve algorithms that solve scheduling and planning problems by using learned representations that understand equivalent MILP formulations.
- •For machine learning developers: Provide pretrained embeddings for mixed-integer programs that speed up downstream prediction tasks without labels from solvers.
Authors
Ousema Bouaneni, Mathis Le Bail, Clément Elliker, Maël Jenny, Sonia Vanier
Abstract
Mixed-integer linear programs (MILP) model many real-world decision problems, motivating machine-learning methods that exploit recurring structure to accelerate MILP solving. MILPs can admit many equivalent formulations: integrality-preserving changes of variables and the addition of redundant constraints can alter their formulations while preserving the optimization problem. We leverage these reformulations as a source of self-supervision for learning general-purpose representations of MILP variables and constraints. We characterize the affine reformulations that are valid for every input instance, and distinguish re-descriptions, which leave variables unchanged, from substitutions, which transform them predictably. Building on equivariant self-supervised learning, we introduce ReMILP (reformulation-contrastive MILP representation learning), which jointly trains a graph neural network and a hypernetwork to predict how variable embeddings transform under changes of variables. Without solver-derived labels, ReMILP learns representations that exhibit the intended invariance and equivariance on unseen problem classes. Across binary solution, constraint activity and integrality gap prediction, these representations carry task-relevant information when frozen and provide a useful initialization for fine-tuning.