# End-to-End Learning to Warm-Start for Real-Time Quadratic Optimization

arXiv: https://arxiv.org/abs/2212.08260 · HTML: https://mlsanity.com/paper/2212.08260/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2022.

## Built on by (most cited first, 26 of 26)

- [Learning to Warm-Start Fixed-Point Optimization Algorithms](https://mlsanity.com/paper/2309.07835/) (2023)
- [DiffuSolve: Diffusion-based Solver for Non-convex Trajectory Optimization](https://mlsanity.com/paper/2403.05571/) (2024)
- [Learning Constrained Optimization with Deep Augmented Lagrangian Methods](https://mlsanity.com/paper/2403.03454/) (2024)
- [Expressive Power of Graph Neural Networks for (Mixed-Integer) Quadratic Programs](https://mlsanity.com/paper/2406.05938/) (2024)
- [Amortized Global Search for Efficient Preliminary Trajectory Design with Deep Generative Models](https://mlsanity.com/paper/2308.03960/) (2023)
- [Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guarantees](https://mlsanity.com/paper/2410.11061/) (2024)
- [Metric Learning to Accelerate Convergence of Operator Splitting Methods for Differentiable Parametric Programming](https://mlsanity.com/paper/2404.00882/) (2024)
- [AC4MPC: Actor-Critic Reinforcement Learning for Nonlinear Model Predictive Control](https://mlsanity.com/paper/2406.03995/) (2024)
- [Data-Driven Performance Guarantees for Classical and Learned Optimizers](https://mlsanity.com/paper/2404.13831/) (2024)
- [Predict-Then-Optimize by Proxy: Learning Joint Models of Prediction and Optimization](https://mlsanity.com/paper/2311.13087/) (2023)
- [Self-Supervised Learning of Iterative Solvers for Constrained Optimization](https://mlsanity.com/paper/2409.08066/) (2024)
- [Towards graph neural networks for provably solving convex optimization problems](https://mlsanity.com/paper/2502.02446/) (2025)
- [Deep FlexQP: Accelerated Nonlinear Programming via Deep Unfolding](https://mlsanity.com/paper/2512.01565/) (2025)
- [LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layers](https://mlsanity.com/paper/2604.05374/) (2026)
- [Learning Joint Models of Prediction and Optimization](https://mlsanity.com/paper/2409.04898/) (2024)
- [Warm-starting active-set solvers using graph neural networks](https://mlsanity.com/paper/2511.13174/) (2025)
- [Scalable Deep Unfolding of Conic Optimizers](https://mlsanity.com/paper/2606.13825/) (2026)
- [Learning to Relax: Setting Solver Parameters Across a Sequence of Linear System Instances](https://mlsanity.com/paper/2310.02246/) (2023)
- [Predicting Accurate Lagrangian Multipliers for Mixed Integer Linear Programs](https://mlsanity.com/paper/2310.14659/) (2023)
- [Learning to Solve Constrained Bilevel Control Co-Design Problems](https://mlsanity.com/paper/2507.09050/) (2025)
- [Reinforcement learning for adaptive interior point methods in convex quadratic programming](https://mlsanity.com/paper/2509.07404/) (2025)
- [Data-driven Projection Generation for Efficiently Solving Heterogeneous Quadratic Programming Problems](https://mlsanity.com/paper/2510.26061/) (2025)
- [UBP2: Uncertainty-Balanced Preference Planning for Efficient Preference-based Reinforcement Learning](https://mlsanity.com/paper/2606.19328/) (2026)
- [Memory-Computation Tradeoffs in Semi Amortized Parametric Optimization](https://mlsanity.com/paper/2607.20769/) (2026)
- [Constrained Graph Diffusion for Mixed Integer Optimization](https://mlsanity.com/paper/2608.13079/) (2026)
- [A Scalable Learned Iterative Solver for Nonlinear MPC via Temporal Decomposition](https://mlsanity.com/paper/2610.13480/) (2026)

## Abstract (arXiv)

First-order methods are widely used to solve convex quadratic programs (QPs) in real-time applications because of their low per-iteration cost. However, they can suffer from slow convergence to accurate solutions. In this paper, we present a framework which learns an effective warm-start for a popular first-order method in real-time applications, Douglas-Rachford (DR) splitting, across a family of parametric QPs. This framework consists of two modules: a feedforward neural network block, which takes as input the parameters of the QP and outputs a warm-start, and a block which performs a fixed number of iterations of DR splitting from this warm-start and outputs a candidate solution. A key feature of our framework is its ability to do end-to-end learning as we differentiate through the DR iterations. To illustrate the effectiveness of our method, we provide generalization bounds (based on Rademacher complexity) that improve with the number of training problems and number of iterations simultaneously. We further apply our method to three real-time applications and observe that, by learning good warm-starts, we are able to significantly reduce the number of iterations required to obtain high-quality solutions.