发表机构
Bellini College of Artificial Intelligence, Cybersecurity and Computing, University of South Florida; Muma College of Business, University of South Florida; University of South Florida（南佛罗里达大学贝利尼人工智能、网络安全与计算学院; 南佛罗里达大学穆玛商学院; 南佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出Graph4BiLO，一种基于GNN的双层值函数学习方法，在背包拦截实例上与MibS、Neur2BiLO对比，取得相当目标值且可零样本跨规模迁移，但存在嵌入消息传递导致公式规模与求解时间增加的权衡问题。
AI 中文摘要
双层混合整数线性优化问题可对分层决策过程进行建模，其中领导者需预测追随者的最优响应。尽管这类问题表达能力强，但由于下层最优性被嵌入到领导者的可行域中，其计算极具挑战性。值函数重构方法用涉及追随者最优值的约束替代嵌套的追随者优化过程，但精确计算该值函数本身可能代价高昂。本文提出Graph4BiLO，一种基于变量-约束图表示学习双层值函数的图神经网络（GNN）方法。与固定长度的多层感知器（MLP）表示不同，GNN使用共享的消息传递参数，因此单个训练好的模型可应用于多种问题规模。学习到的ReLU网络被精确编码为混合整数线性约束，并嵌入到近似的单层公式中。随后的修复步骤会针对选定的领导者决策重新求解追随者问题，以恢复符合双层可行性的追随者响应。我们在包含20至100个物品的背包拦截实例上，将Graph4BiLO与精确求解器MibS及基于学习的方法Neur2BiLO进行对比评估。在所有测试规模下，Graph4BiLO取得的目标值与Neur2BiLO相当，同时避免了使用特定规模的神经网络。额外的分布外实验显示，其能从20个物品的训练实例零样本迁移到此前未见过的40和60个物品实例。不过，在每个图节点嵌入消息传递会大幅增加生成的混合整数公式规模和求解时间。这些结果明确了可泛化规模的图表示与在优化模型中嵌入GNN的计算成本之间的核心权衡。
英文摘要
Bilevel mixed-integer linear optimization problems model hierarchical decision processes in which a leader anticipates the optimal response of a follower. Although expressive, these problems are computationally challenging because lower-level optimality is embedded in the leader's feasible region. Value-function reformulations replace the nested follower optimization with a constraint involving the follower's optimal value, but evaluating this value function exactly can itself be expensive. This paper introduces Graph4BiLO, a graph neural network (GNN) approach for learning bilevel value functions from variable--constraint graph representations. In contrast to fixed-length multilayer perceptron (MLP) representations, the GNN uses shared message-passing parameters and can therefore be applied across multiple problem sizes with a single trained model. The learned ReLU network is encoded exactly as mixed-integer linear constraints and embedded in an approximate single-level formulation. A repair step subsequently re-solves the follower problem for the selected leader decision to recover a bilevel-feasible follower response. We evaluate Graph4BiLO on knapsack interdiction instances with 20--100 items against the exact MibS solver and the learning-based Neur2BiLO method. Graph4BiLO obtains objective values comparable to Neur2BiLO across all tested sizes while avoiding size-specific neural networks. An additional out-of-distribution experiment demonstrates zero-shot transfer from 20-item training instances to previously unseen 40- and 60-item instances. However, embedding message passing at every graph node substantially increases the resulting mixed-integer formulation size and solve time. These results identify a central tradeoff between size-generalizable graph representations and the computational cost of embedding GNNs within optimization models.
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