Recently, machine learning, particularly message-passing graph neural networks (MPNNs), has gained traction in enhancing exact optimization algorithms. For example, MPNNs speed up solving mixed-integer optimization problems by imitating computational intensive heuristics like strong branching, which entails solving multiple linear optimization problems (LPs). Despite the empirical success, the reasons behind MPNNs' effectiveness in emulating linear optimization remain largely unclear. Here, we show that MPNNs can simulate standard interior-point methods for LPs, explaining their practical success. Furthermore, we highlight how MPNNs can serve as a lightweight proxy for solving LPs, adapting to a given problem instance distribution. Empirically, we show that MPNNs solve LP relaxations of standard combinatorial optimization problems close to optimality, often surpassing conventional solvers and competing approaches in solving time.

翻译：近期，机器学习（特别是消息传递图神经网络）在增强精确优化算法方面备受关注。例如，MPNN通过模仿计算密集型启发式方法（如强分支法，该方法需求解多个线性优化问题）来加速混合整数优化问题的求解。尽管取得了实证成功，MPNN在模拟线性优化过程中展现有效性的根本原因仍不清楚。本文证明，MPNN能够模拟求解线性规划的标准内点法，从而解释了其实践有效性。此外，我们揭示了MPNN如何作为求解线性规划的轻量级代理，并适应特定问题实例的分布特征。实验表明，MPNN在标准组合优化问题的线性规划松弛求解中接近最优解，其求解时间往往优于传统求解器及竞争方法。