# Deep FlexQP: Accelerated Nonlinear Programming via Deep Unfolding

arXiv: https://arxiv.org/abs/2512.01565 · HTML: https://mlsanity.com/paper/2512.01565/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 118 references that are in the map.
- First posted on arXiv 1 Dec 2025.

## Builds on (most cited first, 30 of 44)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [Model-Based Deep Learning](https://mlsanity.com/paper/2012.08405/) (2020)
- [A predictive safety filter for learning-based control of constrained nonlinear dynamical systems](https://mlsanity.com/paper/1812.05506/) (2018)
- [Understanding and correcting pathologies in the training of learned optimizers](https://mlsanity.com/paper/1810.10180/) (2018)
- [Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds](https://mlsanity.com/paper/1808.10038/) (2018)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [Deep Unfolding: Model-Based Inspiration of Novel Deep Architectures](https://mlsanity.com/paper/1409.2574/) (2014)
- [CNN-Based Projected Gradient Descent for Consistent Image Reconstruction](https://mlsanity.com/paper/1709.01809/) (2017)
- [End-to-End Text-Dependent Speaker Verification](https://mlsanity.com/paper/1509.08062/) (2015)
- [Neural Proximal Gradient Descent for Compressive Imaging](https://mlsanity.com/paper/1806.03963/) (2018)
- [Deep Unfolded Robust PCA with Application to Clutter Suppression in Ultrasound](https://mlsanity.com/paper/1811.08252/) (2018)
- [Learning to Warm-Start Fixed-Point Optimization Algorithms](https://mlsanity.com/paper/2309.07835/) (2023)
- [Accelerating Quadratic Optimization with Reinforcement Learning](https://mlsanity.com/paper/2107.10847/) (2021)
- [Large Scale Model Predictive Control with Neural Networks and Primal Active Sets](https://mlsanity.com/paper/1910.10835/) (2019)
- [End-to-End Learning to Warm-Start for Real-Time Quadratic Optimization](https://mlsanity.com/paper/2212.08260/) (2022)
- [Safeguarded Learned Convex Optimization](https://mlsanity.com/paper/2003.01880/) (2020)
- [PAC-Bayes Control: Learning Policies that Provably Generalize to Novel Environments](https://mlsanity.com/paper/1806.04225/) (2018)
- [Hyperparameter Tuning is All You Need for LISTA](https://mlsanity.com/paper/2110.15900/) (2021)
- [Multi-Period Trading via Convex Optimization](https://mlsanity.com/paper/1705.00109/) (2017)
- [The Curse of Unrolling: Rate of Differentiating Through Optimization](https://mlsanity.com/paper/2209.13271/) (2022)
- [Towards Constituting Mathematical Structures for Learning to Optimize](https://mlsanity.com/paper/2305.18577/) (2023)
- [Understanding Trainable Sparse Coding via Matrix Factorization](https://mlsanity.com/paper/1609.00285/) (2016)
- [Accelerated Forward-Backward Optimization using Deep Learning](https://mlsanity.com/paper/2105.05210/) (2021)
- [Deep Distributed Optimization for Large-Scale Quadratic Programming](https://mlsanity.com/paper/2412.12156/) (2024)
- [Generalized Kalman Smoothing: Modeling and Algorithms](https://mlsanity.com/paper/1609.06369/) (2016)

## Built on by (most cited first, 4 of 4)

- [Scalable Deep Unfolding of Conic Optimizers](https://mlsanity.com/paper/2606.13825/) (2026)
- [Parametric Nonconvex Optimization via Convex Surrogates](https://mlsanity.com/paper/2604.05640/) (2026)
- [When Descent Is Too Stable: Event-Triggered Hamiltonian Learning to Optimize](https://mlsanity.com/paper/2605.06868/) (2026)
- [Reinforcement Learning to Accelerate Primal-Dual Hybrid Gradient for Linear Programming](https://mlsanity.com/paper/2610.01546/) (2026)

## Abstract (arXiv)

We propose FlexQP, an always-feasible convex quadratic programming (QP) solver based on an ℓ1 elastic relaxation of the QP constraints. If the original constraints are feasible, FlexQP provably recovers the optimal solution. If the constraints are infeasible, FlexQP identifies a solution that minimizes the constraint violation while keeping the number of violated constraints sparse. Such infeasibilities arise naturally in sequential quadratic programming (SQP) subproblems due to the linearization of the constraints. We prove the convergence of FlexQP under mild coercivity assumptions, making it robust to both feasible and infeasible QPs. We then apply deep unfolding to learn LSTM-based, dimension-agnostic feedback policies for the algorithm parameters, yielding an accelerated Deep FlexQP. To preserve the exactness guarantees of the relaxation, we propose a normalized training loss that incorporates the Lagrange multipliers. We additionally design a log-scaled loss for PAC-Bayes generalization bounds that yields substantially tighter performance certificates, which we use to construct an accelerated SQP solver with guaranteed QP subproblem performance. Deep FlexQP outperforms state-of-the-art learned QP solvers on a suite of benchmarks including portfolio optimization, classification, and regression problems, and scales to dense QPs with over 10k variables and constraints via fine-tuning. When deployed within SQP, our approach solves nonlinear trajectory optimization problems 4-16x faster than SQP with OSQP while substantially improving success rates. On predictive safety filter problems, Deep FlexQP reduces safety violations by over 70% and increases task completion by 43% compared to existing methods.