AT-SKM-Net：一种用于动态图上线性硬约束可行性的加速可训练采样Kaczmarz-Motzkin框架
AT-SKM-Net: An Accelerated Trainable Sampling Kaczmarz-Motzkin Framework for Linear Hard-Constraint Feasibility on Dynamic Graphs
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- Zhejiang University（浙江大学）
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中文总结 AI 辅助
提出AT-SKM-Net框架，通过混合采样与Cholesky更新，在动态图上实现线性硬约束可行性求解，迭代减少85%，加速2.95-7.29倍，零违规。
中文摘要 AI 辅助
具有线性约束的图结构优化对关键基础设施至关重要，但由于大量严格的硬约束和高维度，面临可扩展性限制。尽管最近的基于投影的方法（如可训练采样Kaczmarz-Motzkin网络（T-SKM-Net））保证了可行性，但它们在动态环境中处理整个约束集并需要昂贵的矩阵分解，导致高计算成本。为弥合这一差距，我们提出了加速可训练SKM（AT-SKM）网络框架。为了将计算集中在活动约束上并消除冗余计算，我们引入了一种由拓扑感知的异构GNN模型引导的混合采样策略。为了有效处理基于图的约束中的拓扑变化，我们采用Cholesky更新机制，该机制在理论上将低秩扰动下的等式投影复杂度从O(N^3)降低到O(N^2)。在随机几何图、N-1安全约束直流最优潮流和最小成本天然气输送问题上的实验表明，AT-SKM将迭代次数减少高达85%，并实现2.95倍至7.29倍的SKM层加速，同时保持零约束违规。
英文摘要
Graph-structured optimization with linear constraints is fundamental to critical infrastructure but faces scalability limits due to massive strict hard constraints and high dimensionality. While recent projection-based methods such as Trainable Sampling Kaczmarz-Motzkin Net (T-SKM-Net) guarantee feasibility, they face high computational costs in dynamic environments by processing the entire constraint set and requiring expensive matrix factorizations. To bridge this gap, we propose the Accelerated Trainable-SKM (AT-SKM) Net framework. To concentrate computation on the active constraints and eliminate redundant calculations, we introduce a hybrid sampling strategy guided by a topology-aware heterogeneous GNN model. To efficiently handle topological shifts in graph-based constraints, we employ a Cholesky Update mechanism that theoretically reduces the equality projection complexity from O(N^3) to O(N^2) under low-rank perturbations. Experiments on random geometric graphs, N-1 Security-Constrained DC-OPF, and minimum-cost gas transport problem demonstrate that AT-SKM reduces iteration counts by up to 85% and achieves 2.95x-7.29x SKM layer speedups, while maintaining zero constraint violations.
