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

arXiv: https://arxiv.org/abs/2609.30088 · HTML: https://mlsanity.com/paper/2609.30088/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 43 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 25 of 25)

- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [DeepOPF: A Deep Neural Network Approach for Security-Constrained DC Optimal Power Flow](https://mlsanity.com/paper/1910.14448/) (2019)
- [Constrained Learning with Non-Convex Losses](https://mlsanity.com/paper/2103.05134/) (2021)
- [Learning for Constrained Optimization: Identifying Optimal Active Constraint Sets](https://mlsanity.com/paper/1802.09639/) (2018)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [Learning Constrained Optimization with Deep Augmented Lagrangian Methods](https://mlsanity.com/paper/2403.03454/) (2024)
- [LinSATNet: The Positive Linear Satisfiability Neural Networks](https://mlsanity.com/paper/2407.13917/) (2024)
- [Screening Tests for Lasso Problems](https://mlsanity.com/paper/1405.4897/) (2014)
- [Statistical Learning For DC Optimal Power Flow](https://mlsanity.com/paper/1801.07809/) (2018)
- [A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraints](https://mlsanity.com/paper/2307.10459/) (2023)
- [Enforcing Hard Linear Constraints in Deep Learning Models with Decision Rules](https://mlsanity.com/paper/2505.13858/) (2025)
- [A Sampling Kaczmarz-Motzkin Algorithm for Linear Feasibility](https://mlsanity.com/paper/1605.01418/) (2016)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [Sampling Kaczmarz Motzkin Method for Linear Feasibility Problems: Generalization & Acceleration](https://mlsanity.com/paper/2002.07321/) (2020)
- [GRANOLA: Adaptive Normalization for Graph Neural Networks](https://mlsanity.com/paper/2404.13344/) (2024)
- [GLinSAT: The General Linear Satisfiability Neural Network Layer By Accelerated Gradient Descent](https://mlsanity.com/paper/2409.17500/) (2024)
- [SCQPTH: an efficient differentiable splitting method for convex quadratic programming](https://mlsanity.com/paper/2308.08232/) (2023)
- [Enforcing convex constraints in Graph Neural Networks](https://mlsanity.com/paper/2510.11227/) (2025)
- [Warm-starting active-set solvers using graph neural networks](https://mlsanity.com/paper/2511.13174/) (2025)
- [T-SKM-Net: Trainable Neural Network Framework for Linear Constraint Satisfaction via Sampling Kaczmarz-Motzkin Method](https://mlsanity.com/paper/2512.10461/) (2025)

## Abstract (arXiv)

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.