迈向可证明求解凸优化问题的图神经网络
Towards graph neural networks for provably solving convex optimization problems
AI总结：
本文提出一种迭代式消息传递图神经网络框架，可证明地模拟内点法并保证可行性，在凸优化问题求解质量、泛化性和速度上优于现有神经基线及部分商业求解器。
摘要：
近年来，消息传递图神经网络（MPNNs）因其能够捕捉变量与约束之间的交互关系，在求解组合优化和连续优化问题方面展现出潜力。尽管现有方法利用MPNNs来近似求解或为传统求解器提供热启动，但它们往往缺乏可行性保证，尤其是在凸优化场景中。在此，我们提出一种迭代式MPNN框架，用于求解具有可证明可行性保证的凸优化问题。首先，我们证明MPNNs能够可证明地模拟标准内点法，用于求解带线性约束的二次问题，涵盖支持向量机（SVMs）等相关问题。其次，为确保可行性，我们引入一种变体，该变体从可行点出发，并在可行区域内迭代地限制搜索范围。实验结果表明，我们的方法在解的质量和可行性方面优于现有神经基线，能够很好地泛化到未见过的问题规模，并且在某些情况下，比Gurobi等最先进的求解器获得更快的求解时间。
英文摘要：
Recently, message-passing graph neural networks (MPNNs) have shown potential for solving combinatorial and continuous optimization problems due to their ability to capture variable-constraint interactions. While existing approaches leverage MPNNs to approximate solutions or warm-start traditional solvers, they often lack guarantees for feasibility, particularly in convex optimization settings. Here, we propose an iterative MPNN framework to solve convex optimization problems with provable feasibility guarantees. First, we demonstrate that MPNNs can provably simulate standard interior-point methods for solving quadratic problems with linear constraints, covering relevant problems such as SVMs. Secondly, to ensure feasibility, we introduce a variant that starts from a feasible point and iteratively restricts the search within the feasible region. Experimental results show that our approach outperforms existing neural baselines in solution quality and feasibility, generalizes well to unseen problem sizes, and, in some cases, achieves faster solution times than state-of-the-art solvers such as Gurobi.
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