ML-Guided Primal Heuristics for Mixed Binary Quadratic Programs

2026-04-24

AI Summary

This work addresses the challenge of solving mixed binary quadratic programming (MBQP) problems, which are notoriously difficult and for which existing heuristics often fail to produce high-quality feasible solutions within limited time. The authors propose a machine learning–based primal heuristic featuring a novel neural network architecture tailored for MBQP, coupled with an efficient training data collection pipeline. To enhance model generalization, they integrate contrastive loss with weighted cross-entropy loss and introduce a cross-regime transfer inference mechanism. Extensive evaluations on both standard and real-world MBQP benchmarks demonstrate that the proposed method significantly outperforms state-of-the-art heuristics and commercial solvers. Furthermore, it exhibits strong generalization capabilities in practical applications, notably in wind farm layout optimization.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: ML for personalized search and recommendations

Abstract

Mixed Binary Quadratic Programs (MBQPs) are an important and complex set of problems in combinatorial optimization. As solving large-scale combinatorial optimization problems is challenging, primal heuristics have been developed to quickly identify high-quality solutions within a short amount of time. Recently, a growing body of research has also used machine learning to accelerate solution methods for challenging combinatorial optimization problems. Despite the increasing popularity of these ML-guided methods, a large body of work has focused on Mixed-Integer Linear Programs (MILPs). MBQPs are challenging to solve due to the combinatorial complexity coupled with nonlinearities. This work proposes ML-guided primal heuristics for Mixed Binary Quadratic Programs (MBQPs) by adapting and extending existing work on ML-guided MILP solution prediction to MBQPs. We introduce a new neural network architecture for MBQP solution prediction and a new training data collection procedure. Moreover, we extend existing loss functions in solution prediction and propose to combine contrastive and weighted cross-entropy losses. We evaluate the methods on standard and real-world MBQP benchmarks and show that the developed ML-guided methods significantly outperform existing primal heuristics and state-of-the-art solvers. Furthermore, models trained with our proposed extension with combined losses outperform other ML-based methods adapted from MILPs and improve generalization in cross-regional inference on a real-world wind farm layout optimization problem.

Problem

• Mixed Binary Quadratic Programs
• Combinatorial Optimization
• Primal Heuristics
• Machine Learning
• Nonlinear Optimization

Innovation

• ML-guided primal heuristics
• Mixed Binary Quadratic Programs
• neural network architecture
• contrastive loss
• solution prediction