AT-SKM-Net: An Accelerated Trainable Sampling Kaczmarz-Motzkin Framework for Linear Hard-Constraint Feasibility on Dynamic Graphs

2026-09-24

AI Summary

This study addresses the high computational complexity and limited scalability of traditional projection methods in linear hard-constrained optimization over dynamic graphs, which stem from full constraint processing and matrix decomposition. We propose a topology-aware, accelerated, and trainable Sketching Kaczmarz Method (SKM) framework. This approach leverages heterogeneous graph neural networks to guide a hybrid sampling strategy that focuses on active constraints, while incorporating a Cholesky update mechanism to efficiently accommodate graph topology changes, thereby reducing equality projection complexity from O(N³) to O(N²). Experimental results demonstrate that the proposed framework strictly maintains zero constraint violation while reducing the number of iterations by 85% and achieving a 2.95× to 7.29× inference speedup for the SKM layer.

Abstract

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.

Problem

• Graph-structured optimization
• Linear hard constraints
• Dynamic graphs
• Scalability
• Computational cost

Innovation

• Trainable Sampling Kaczmarz-Motzkin
• Topology-aware Heterogeneous GNN
• Cholesky Update
• Dynamic Graphs
• Linear Hard Constraints