# Efficient differentiable quadratic programming layers: an ADMM approach

arXiv: https://arxiv.org/abs/2112.07464 · HTML: https://mlsanity.com/paper/2112.07464/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 51 references that are in the map.
- First posted on arXiv 14 Dec 2021.

## Builds on (most cited first, 22 of 22)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [From Predictive to Prescriptive Analytics](https://mlsanity.com/paper/1402.5481/) (2014)
- [Smart Predict-and-Optimize for Hard Combinatorial Optimization Problems](https://mlsanity.com/paper/1911.10092/) (2019)
- [Interior Point Solving for LP-based prediction+optimisation](https://mlsanity.com/paper/2010.13943/) (2020)
- [Stochastic Hyperparameter Optimization through Hypernetworks](https://mlsanity.com/paper/1802.09419/) (2018)
- [Differentiating Through a Cone Program](https://mlsanity.com/paper/1904.09043/) (2019)
- [Decision Trees for Decision-Making under the Predict-then-Optimize Framework](https://mlsanity.com/paper/2003.00360/) (2020)
- [Unrolled Optimization with Deep Priors](https://mlsanity.com/paper/1705.08041/) (2017)
- [End-to-End Learning for Structured Prediction Energy Networks](https://mlsanity.com/paper/1703.05667/) (2017)
- [ADMM-Net: A Deep Learning Approach for Compressive Sensing MRI](https://mlsanity.com/paper/1705.06869/) (2017)
- [Solution Refinement at Regular Points of Conic Problems](https://mlsanity.com/paper/1811.02157/) (2018)
- [Differentiable Linearized ADMM](https://mlsanity.com/paper/1905.06179/) (2019)
- [GPU Acceleration of ADMM for Large-Scale Quadratic Programming](https://mlsanity.com/paper/1912.04263/) (2019)
- [Task-based End-to-end Model Learning in Stochastic Optimization](https://mlsanity.com/paper/1703.04529/) (2017)
- [Gradient-based Regularization Parameter Selection for Problems with Non-smooth Penalty Functions](https://mlsanity.com/paper/1703.09813/) (2017)
- [SuperSCS: fast and accurate large-scale conic optimization](https://mlsanity.com/paper/1903.06477/) (2019)

## Built on by (most cited first, 19 of 19)

- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Alternating Differentiation for Optimization Layers](https://mlsanity.com/paper/2210.01802/) (2022)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [End-to-end Conditional Robust Optimization](https://mlsanity.com/paper/2403.04670/) (2024)
- [Differentiation Through Black-Box Quadratic Programming Solvers](https://mlsanity.com/paper/2410.06324/) (2024)
- [BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning](https://mlsanity.com/paper/2411.19285/) (2024)
- [Feasibility-Aware Decision-Focused Learning for Predicting Parameters in the Constraints](https://mlsanity.com/paper/2510.04951/) (2025)
- [SCQPTH: an efficient differentiable splitting method for convex quadratic programming](https://mlsanity.com/paper/2308.08232/) (2023)
- [A Fully First-Order Layer for Differentiable Optimization](https://mlsanity.com/paper/2512.02494/) (2025)
- [Smart Surrogate Losses for Contextual Stochastic Linear Optimization with Robust Constraints](https://mlsanity.com/paper/2505.22881/) (2025)
- [DiffCoord: Differentiable Coordination for Distributed Multi-Agent Trajectory Optimization](https://mlsanity.com/paper/2509.01630/) (2025)
- [A Penalty Approach for Differentiation Through Black-Box Quadratic Programming Solvers](https://mlsanity.com/paper/2602.14154/) (2026)
- [Gradient boosting for convex cone predict and optimize problems](https://mlsanity.com/paper/2204.06895/) (2022)
- [Learning Deterministic Surrogates for Robust Convex QCQPs](https://mlsanity.com/paper/2312.12485/) (2023)
- [T-SKM-Net: Trainable Neural Network Framework for Linear Constraint Satisfaction via Sampling Kaczmarz-Motzkin Method](https://mlsanity.com/paper/2512.10461/) (2025)
- [QuadAttack: A Quadratic Programming Approach to Ordered Top-K Attacks](https://mlsanity.com/paper/2312.11510/) (2023)
- [Data-driven Projection Generation for Efficiently Solving Heterogeneous Quadratic Programming Problems](https://mlsanity.com/paper/2510.26061/) (2025)
- [lapanda: A Matrix-Free Differentiable Solver for Nonconvex Constrained Optimization Layers](https://mlsanity.com/paper/2609.33630/) (2026)
- [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026)

## Abstract (arXiv)

Recent advances in neural-network architecture allow for seamless integration of convex optimization problems as differentiable layers in an end-to-end trainable neural network. Integrating medium and large scale quadratic programs into a deep neural network architecture, however, is challenging as solving quadratic programs exactly by interior-point methods has worst-case cubic complexity in the number of variables. In this paper, we present an alternative network layer architecture based on the alternating direction method of multipliers (ADMM) that is capable of scaling to problems with a moderately large number of variables. Backward differentiation is performed by implicit differentiation of the residual map of a modified fixed-point iteration. Simulated results demonstrate the computational advantage of the ADMM layer, which for medium scaled problems is approximately an order of magnitude faster than the OptNet quadratic programming layer. Furthermore, our novel backward-pass routine is efficient, from both a memory and computation standpoint, in comparison to the standard approach based on unrolled differentiation or implicit differentiation of the KKT optimality conditions. We conclude with examples from portfolio optimization in the integrated prediction and optimization paradigm.