Dynamic Stratified Contrastive Learning with Upstream Augmentation for MILP Branching

TL;DR

An upstream-augmented MILP derivation procedure is introduced that generates both theoretically equivalent and perturbed instances and generalizes effectively to unseen instances, significantly enhancing branching accuracy and solving efficiency, particularly for upstream nodes.

Abstract

Mixed Integer Linear Programming (MILP) is a fundamental class of NP-hard problems that has garnered significant attention from both academia and industry. The Branch-and-Bound (B\&B) method is the dominant approach for solving MILPs and the branching plays an important role in B\&B methods. Neural-based learning frameworks have recently been developed to enhance branching policies and the efficiency of solving MILPs. However, these methods still struggle with semantic variation across depths, the scarcity of upstream nodes, and the costly collection of strong branching samples. To address these issues, we propose \ours, a Dynamic \underline{\textbf{S}}tratified \underline{\textbf{C}}ontrastive Training Framework for \underline{\textbf{MILP}} Branching. It groups branch-and-bound nodes based on their feature distributions and trains a GCNN-based discriminative model to progressively separate nodes across groups, learning finer-grained node representations throughout the tree. To address data scarcity and imbalance at upstream nodes, we introduce an upstream-augmented MILP derivation procedure that generates both theoretically equivalent and perturbed instances. \ours~effectively models subtle semantic differences between nodes, significantly enhancing branching accuracy and solving efficiency, particularly for upstream nodes. Extensive experiments on standard MILP benchmarks demonstrate that our method enhances branching accuracy, reduces solving time, and generalizes effectively to unseen instances.

Figures (2)

• Figure 1: Overview of our SC-MILP. Each node in the B&B tree is color-coded to reflect feature variations across depths, while nodes marked with slashes denote additional samples generated via upstream-augmented MILP derivation. Nodes are first grouped into feature-driven strata ($G_1, G_2, \dots$). Both equivalent and perturbed MILP derivations are then applied to augment upstream samples. Specifically, LT-MILP and RC-MILP refer to derivations based on linear transformation and redundant constraint generation, respectively, whereas objective Per., constraint Per., and dual variable Per. indicate perturbations applied to the objective, constraints, and dual variables. Finally, dynamic stratified contrastive learning is performed, where positives are defined within the same stratum and negatives across different strata, with dynamic stratified weighting that progressively increases separation with group depth and is adaptively modulated during training.
• Figure 2: Parameter analysis. In each subplot, SC indicates Set Covering, CA indicates Combinatorial Auction, CFL indicates Capacitated Facility Location, and MIS indicates Maximum Independent Set.

Theorems & Definitions (10)

• Definition 4.1: Linear transformation based Equivalent MILP Derivation (LT-MILP)
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4
• Definition 4.5: Redundant Constraint based Equivalent MILP Derivation (RC-MILP)
• Definition 4.6: Objective and Constraint Perturbation
• Definition 4.7: Dual Variable Perturbation
• proof : Proof of Theorem \ref{['theorem1']}
• proof : Proof of Theorem \ref{['theorem2']}
• proof : Proof of Theorem \ref{['theorem3']}