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Lecture notes in computer science

Towards Solving Polynomial-Objective Integer Programming with Hypergraph Neural Networks

Minshuo Li, Yizhi Wu, Pavel Troubil, Yingqian Zhang, Wim Nuijten

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.1007/978-3-032-27242-3_18

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Binary (0-1) integer programming (BIP) is pivotal in scientific domains requiring discrete decision-making. As the advance of AI computing, recent works explore neural network-based solvers for integer linear programming (ILP) problems. Yet, they lack scalability for tackling nonlinear challenges. To handle nonlinearities, state-of-the-art Branch-and-Cut solvers employ linear relaxations, leading to exponential growth in auxiliary variables and severe computation limitations. To overcome these limitations, we propose BIPNN (Binary Integer Programming Neural Network), an unsupervised learning framework to solve nonlinear BIP problems via hypergraph neural networks (HyperGNN). Specifically, BIPNN reformulates BIPs-constrained, discrete, and nonlinear (sin, log, exp) optimization problems-into unconstrained, differentiable, and polynomial loss functions. The reformulation stems from the observation of a precise one-to-one mapping between polynomial BIP objectives and hypergraph structures, enabling the unsupervised training of HyperGNN to optimize BIP problems in an end-to-end manner. On this basis, we propose a GPU-accelerated and continuous-annealing-enhanced training pipeline for BIPNN. The pipeline enables BIPNN to optimize large-scale nonlinear terms in BIPs fully in parallel via straightforward gradient descent, thus significantly reducing the training cost while ensuring the generation of discrete, high-quality solutions. Extensive experiments on synthetic and real-world datasets highlight the superiority of our approach.
Machine learning-based optimization frameworks have attracted increasing attention for accelerating the solution of large-scale quadratically constrained quadratic programs (QCQPs) by exploiting shared problem structure across instances. However, existing machine learning (ML) frameworks often rely on the assumption of parametric models and large-scale solvers. This article introduces HEQP, a hypergraph neural network-based evolutionary optimization framework for large-scale QCQPs. This framework features two main components: 1) hypergraph-based neural prediction, which predicts optimal solutions for QCQPs without assumptions of models; and 2) evolutionary large neighborhood search (Evo-LNS), which employs a McCormick relaxation-based repair strategy to search and apply crossover on neighborhood solutions using a small-scale solver. We further show that our framework is equivalent to the interior-point method (IPM), a polynomial-time algorithm, for quadratic programming. Experiments on two types of benchmark problems and 13 large-scale real-world instances from the QPLIB illustrate that our framework outperforms state-of-the-art solvers (including Gurobi, SCIP, and SHOT) in both solution quality and time efficiency, highlighting the efficiency of ML-based optimization frameworks for QCQPs.