CoCo-MILP: Inter-Variable Contrastive and Intra-Constraint Competitive MILP Solution Prediction

TL;DR

CoCo-MILP addresses two core misalignments in learning-based MILP solvers: independent binary predictions under BCE and the smoothing tendency of standard GNNs that blurs intra-constraint competition. It introduces Inter-Variable Contrastive Loss (VCL) to directly optimize relative variable ordering and an Intra-Constraint Competitive (ICC) GNN layer to differentiate competing variables within a constraint. Empirical results on synthetic benchmarks and MIPLIB show substantial primal-gap reductions and instances achieving or matching the best-known solutions, often outperforming strong baselines like PS, ConPS, and Apollo-MILP. The approach demonstrates strong generalization and compatibility as a plug-in predictor that enhances traditional solvers such as Gurobi and SCIP, with practical implications for faster, higher-quality MILP solving in real-world applications.

Abstract

Mixed-Integer Linear Programming (MILP) is a cornerstone of combinatorial optimization, yet solving large-scale instances remains a significant computational challenge. Recently, Graph Neural Networks (GNNs) have shown promise in accelerating MILP solvers by predicting high-quality solutions. However, we identify that existing methods misalign with the intrinsic structure of MILP problems at two levels. At the leaning objective level, the Binary Cross-Entropy (BCE) loss treats variables independently, neglecting their relative priority and yielding plausible logits. At the model architecture level, standard GNN message passing inherently smooths the representations across variables, missing the natural competitive relationships within constraints. To address these challenges, we propose CoCo-MILP, which explicitly models inter-variable Contrast and intra-constraint Competition for advanced MILP solution prediction. At the objective level, CoCo-MILP introduces the Inter-Variable Contrastive Loss (VCL), which explicitly maximizes the embedding margin between variables assigned one versus zero. At the architectural level, we design an Intra-Constraint Competitive GNN layer that, instead of homogenizing features, learns to differentiate representations of competing variables within a constraint, capturing their exclusionary nature. Experimental results on standard benchmarks demonstrate that CoCo-MILP significantly outperforms existing learning-based approaches, reducing the solution gap by up to 68.12% compared to traditional solvers. Our code is available at https://github.com/happypu326/CoCo-MILP.

Figures (7)

• Figure 1: Illustration of CoCo-MILP. We aim to improve the quality of the predicted variables for MILP. The main contribution of our work is the proposed inter-variable contrastive loss and intra-constraint competitive layer for the GNN predictor in the predict-and-search framework.
• Figure 2: Distribution of Logits from different methods.
• Figure 3: Distribution of the difference between logits corresponding to ground truth 1 and 0 from different methods.
• Figure 4: Ratio of variables with value $1$ from different problems. For each problem category, we traverse through all constraints to count the number of variables taking the value 1 in the ground truth and plot the resulting histogram.
• Figure 5: Distribution of logit variance per constraint. IntraVarMean first calculates the variance of logits within each constraint and then computes their average; InterVar calculates the variance across all logits; Ratio represents the ratio of IntraVarMean to InterVar.