# An Efficient Unsupervised Framework for Convex Quadratic Programs via Deep Unrolling

arXiv: https://arxiv.org/abs/2412.01051 · HTML: https://mlsanity.com/paper/2412.01051/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 2 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Solving Mixed Integer Programs Using Neural Networks](https://mlsanity.com/paper/2012.13349/) (2020)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [FISTA-Net: Learning A Fast Iterative Shrinkage Thresholding Network for Inverse Problems in Imaging](https://mlsanity.com/paper/2008.02683/) (2020)
- [The Machine Learning for Combinatorial Optimization Competition (ML4CO): Results and Insights](https://mlsanity.com/paper/2203.02433/) (2022)
- [On Representing Linear Programs by Graph Neural Networks](https://mlsanity.com/paper/2209.12288/) (2022)
- [Multi-Period Trading via Convex Optimization](https://mlsanity.com/paper/1705.00109/) (2017)
- [Operator splitting for a homogeneous embedding of the linear complementarity problem](https://mlsanity.com/paper/2004.02177/) (2020)
- [PDHG-Unrolled Learning-to-Optimize Method for Large-Scale Linear Programming](https://mlsanity.com/paper/2406.01908/) (2024)
- [Expressive Power of Graph Neural Networks for (Mixed-Integer) Quadratic Programs](https://mlsanity.com/paper/2406.05938/) (2024)
- [A Practical and Optimal First-Order Method for Large-Scale Convex Quadratic Programming](https://mlsanity.com/paper/2311.07710/) (2023)
- [Learning to optimize: A tutorial for continuous and mixed-integer optimization](https://mlsanity.com/paper/2405.15251/) (2024)

## Built on by (most cited first, 3 of 3)

- [Unveiling Mode Connectivity in Graph Neural Networks](https://mlsanity.com/paper/2502.12608/) (2025)
- [Principled Data Augmentation for Learning to Solve Quadratic Programming Problems](https://mlsanity.com/paper/2506.01728/) (2025)
- [Machine Learning-Assisted High-Dimensional Matrix Estimation](https://mlsanity.com/paper/2603.28346/) (2026)

## Abstract (arXiv)

Quadratic programs (QPs) arise in various domains such as machine learning, finance, and control. Recently, learning-enhanced primal-dual hybrid gradient (PDHG) methods have shown great potential in addressing large-scale linear programs; however, this approach has not been extended to QPs. In this work, we focus on unrolling "PDQP", a PDHG algorithm specialized for convex QPs. Specifically, we propose a neural network model called "PDQP-net" to learn optimal QP solutions. Theoretically, we demonstrate that a PDQP-net of polynomial size can align with the PDQP algorithm, returning optimal primal-dual solution pairs. We propose an unsupervised method that incorporates KKT conditions into the loss function. Unlike the standard learning-to-optimize framework that requires optimization solutions generated by solvers, our unsupervised method adjusts the network weights directly from the evaluation of the primal-dual gap. This method has two benefits over supervised learning: first, it helps generate better primal-dual gap since the primal-dual gap is in the objective function; second, it does not require solvers. We show that PDQP-net trained in this unsupervised manner can effectively approximate optimal QP solutions. Extensive numerical experiments confirm our findings, indicating that using PDQP-net predictions to warm-start PDQP can achieve up to 45% acceleration on QP instances. Moreover, it achieves 14% to 31% acceleration on out-of-distribution instances.