# Enforcing convex constraints in Graph Neural Networks

arXiv: https://arxiv.org/abs/2510.11227 · HTML: https://mlsanity.com/paper/2510.11227/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 70 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Builds on (most cited first, 30 of 30)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Surrogate Gradient Learning in Spiking Neural Networks](https://mlsanity.com/paper/1901.09948/) (2019)
- [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)
- [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [Optimal Wireless Resource Allocation with Random Edge Graph Neural Networks](https://mlsanity.com/paper/1909.01865/) (2019)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance](https://mlsanity.com/paper/1805.11897/) (2018)
- [Equivariant Descriptor Fields: SE(3)-Equivariant Energy-Based Models for End-to-End Visual Robotic Manipulation Learning](https://mlsanity.com/paper/2206.08321/) (2022)
- [Clarabel: An interior-point solver for conic programs with quadratic objectives](https://mlsanity.com/paper/2405.12762/) (2024)
- [A GNN-Guided Predict-and-Search Framework for Mixed-Integer Linear Programming](https://mlsanity.com/paper/2302.05636/) (2023)
- [On Representing Linear Programs by Graph Neural Networks](https://mlsanity.com/paper/2209.12288/) (2022)
- [Practical Large-Scale Linear Programming using Primal-Dual Hybrid Gradient](https://mlsanity.com/paper/2106.04756/) (2021)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [Learning Permutations with Sinkhorn Policy Gradient](https://mlsanity.com/paper/1805.07010/) (2018)
- [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)
- [Gunrock: A High-Performance Graph Processing Library on the GPU](https://mlsanity.com/paper/1501.05387/) (2015)
- [A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraints](https://mlsanity.com/paper/2307.10459/) (2023)
- [Elucidating the theoretical underpinnings of surrogate gradient learning in spiking neural networks](https://mlsanity.com/paper/2404.14964/) (2024)
- [PIQP: A Proximal Interior-Point Quadratic Programming Solver](https://mlsanity.com/paper/2304.00290/) (2023)
- [Sample-Specific Output Constraints for Neural Networks](https://mlsanity.com/paper/2003.10258/) (2020)
- [Towards graph neural networks for provably solving convex optimization problems](https://mlsanity.com/paper/2502.02446/) (2025)
- [Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review](https://mlsanity.com/paper/1802.07529/) (2018)

## Built on by (most cited first, 2 of 2)

- [Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks](https://mlsanity.com/paper/2605.07113/) (2026)
- [AT-SKM-Net: An Accelerated Trainable Sampling Kaczmarz-Motzkin Framework for Linear Hard-Constraint Feasibility on Dynamic Graphs](https://mlsanity.com/paper/2609.30088/) (2026)

## Abstract (arXiv)

Many machine learning applications require outputs that satisfy complex, dynamic constraints. This task is particularly challenging in Graph Neural Network models due to the variable output sizes of graph-structured data. In this paper, we introduce ProjNet, a Graph Neural Network framework which satisfies input-dependant constraints. ProjNet combines a sparse vector clipping method with the Component-Averaged Dykstra (CAD) algorithm, an iterative scheme for solving the best-approximation problem. We establish a convergence result for CAD and develop a GPU-accelerated implementation capable of handling large-scale inputs efficiently. To enable end-to-end training, we introduce a surrogate gradient for CAD that is both computationally efficient and better suited for optimization than the exact gradient. We validate ProjNet on four classes of constrained optimisation problems: linear programming, two classes of non-convex quadratic programs, and radio transmit power optimization, demonstrating its effectiveness across diverse problem settings.