# PDHG-Unrolled Learning-to-Optimize Method for Large-Scale Linear Programming arXiv: https://arxiv.org/abs/2406.01908 · HTML: https://mlsanity.com/paper/2406.01908/ ## Key facts - Cited by 17 papers in the map (data through 14 Oct 2026). - Builds on 22 of its 50 references that are in the map. - First posted on arXiv 4 Jun 2024. ## Builds on (most cited first, 22 of 22) - [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018) - [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018) - [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019) - [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021) - [Hybrid Models for Learning to Branch](https://mlsanity.com/paper/2006.15212/) (2020) - [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016) - [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) - [Exploring the Power of Graph Neural Networks in Solving Linear Optimization Problems](https://mlsanity.com/paper/2310.10603/) (2023) - [Operator splitting for a homogeneous embedding of the linear complementarity problem](https://mlsanity.com/paper/2004.02177/) (2020) - [Optimized first-order methods for smooth convex minimization](https://mlsanity.com/paper/1406.5468/) (2014) - [Faster First-Order Primal-Dual Methods for Linear Programming using Restarts and Sharpness](https://mlsanity.com/paper/2105.12715/) (2021) - [An ADMM-Based Interior-Point Method for Large-Scale Linear Programming](https://mlsanity.com/paper/1805.12344/) (2018) - [Learning to Pivot as a Smart Expert](https://mlsanity.com/paper/2308.08171/) (2023) - [A Practical and Optimal First-Order Method for Large-Scale Convex Quadratic Programming](https://mlsanity.com/paper/2311.07710/) (2023) - [cuPDLP-C: A Strengthened Implementation of cuPDLP for Linear Programming by C language](https://mlsanity.com/paper/2312.14832/) (2023) - [On a Unified and Simplified Proof for the Ergodic Convergence Rates of PPM, PDHG and ADMM](https://mlsanity.com/paper/2305.02165/) (2023) - [Efficient First-Order Methods for Linear Programming and Semidefinite Programming](https://mlsanity.com/paper/1409.5832/) (2014) - [Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradient](https://mlsanity.com/paper/2206.03041/) (2022) - [On the Infimal Sub-differential Size of Primal-Dual Hybrid Gradient Method and Beyond](https://mlsanity.com/paper/2206.12061/) (2022) - [On the Geometry and Refined Rate of Primal-Dual Hybrid Gradient for Linear Programming](https://mlsanity.com/paper/2307.03664/) (2023) - [Worst-case analysis of restarted primal-dual hybrid gradient on totally unimodular linear programs](https://mlsanity.com/paper/2309.03988/) (2023) ## Built on by (most cited first, 17 of 17) - [Primal-Dual Neural Algorithmic Reasoning](https://mlsanity.com/paper/2505.24067/) (2025) - [Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guarantees](https://mlsanity.com/paper/2410.11061/) (2024) - [Towards graph neural networks for provably solving convex optimization problems](https://mlsanity.com/paper/2502.02446/) (2025) - [An Efficient Unsupervised Framework for Convex Quadratic Programs via Deep Unrolling](https://mlsanity.com/paper/2412.01051/) (2024) - [Principled Data Augmentation for Learning to Solve Quadratic Programming Problems](https://mlsanity.com/paper/2506.01728/) (2025) - [Unrolled Graph Neural Networks for Constrained Optimization](https://mlsanity.com/paper/2509.17156/) (2025) - [Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs](https://mlsanity.com/paper/2602.12273/) (2026) - [On the Expressive Power of GNNs to Solve Linear SDPs](https://mlsanity.com/paper/2604.27786/) (2026) - [Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks](https://mlsanity.com/paper/2605.07113/) (2026) - [Feature Augmentation of GNNs for ILPs: Local Uniqueness Suffices](https://mlsanity.com/paper/2509.21000/) (2025) - [Self-Certifying Primal-Dual Optimization Proxies for Large-Scale Batch Economic Dispatch](https://mlsanity.com/paper/2510.15850/) (2025) - [Unrolled Neural Networks for Constrained Optimization](https://mlsanity.com/paper/2601.17274/) (2026) - [Machine Learning-Assisted High-Dimensional Matrix Estimation](https://mlsanity.com/paper/2603.28346/) (2026) - [FunL2O: LLM-Guided Feature Function Design for Learning to Optimize](https://mlsanity.com/paper/2607.27389/) (2026) - [Reinforcement Learning to Accelerate Primal-Dual Hybrid Gradient for Linear Programming](https://mlsanity.com/paper/2610.01546/) (2026) - [Neural Algorithmic Reasoning for Graph Saddle Point Problems](https://mlsanity.com/paper/2610.07255/) (2026) - [A Scalable Learned Iterative Solver for Nonlinear MPC via Temporal Decomposition](https://mlsanity.com/paper/2610.13480/) (2026) ## Abstract (arXiv) Solving large-scale linear programming (LP) problems is an important task in various areas such as communication networks, power systems, finance and logistics. Recently, two distinct approaches have emerged to expedite LP solving: (i) First-order methods (FOMs); (ii) Learning to optimize (L2O). In this work, we propose an FOM-unrolled neural network (NN) called PDHG-Net, and propose a two-stage L2O method to solve large-scale LP problems. The new architecture PDHG-Net is designed by unrolling the recently emerged PDHG method into a neural network, combined with channel-expansion techniques borrowed from graph neural networks. We prove that the proposed PDHG-Net can recover PDHG algorithm, thus can approximate optimal solutions of LP instances with a polynomial number of neurons. We propose a two-stage inference approach: first use PDHG-Net to generate an approximate solution, and then apply PDHG algorithm to further improve the solution. Experiments show that our approach can significantly accelerate LP solving, achieving up to a 3× speedup compared to FOMs for large-scale LP problems.