# A Neural Network Framework for Discovering Closed-form Solutions to Quadratic Programs with Linear Constraints

arXiv: https://arxiv.org/abs/2510.23737 · HTML: https://mlsanity.com/paper/2510.23737/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 33 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 17 of 17)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Reinforcement Learning for Integer Programming: Learning to Cut](https://mlsanity.com/paper/1906.04859/) (2019)
- [Hybrid Models for Learning to Branch](https://mlsanity.com/paper/2006.15212/) (2020)
- [Predicting AC Optimal Power Flows: Combining Deep Learning and Lagrangian Dual Methods](https://mlsanity.com/paper/1909.10461/) (2019)
- [pandapower - an Open Source Python Tool for Convenient Modeling, Analysis and Optimization of Electric Power Systems](https://mlsanity.com/paper/1709.06743/) (2017)
- [Learning to Branch](https://mlsanity.com/paper/1803.10150/) (2018)
- [Learning Optimal Solutions for Extremely Fast AC Optimal Power Flow](https://mlsanity.com/paper/1910.01213/) (2019)
- [Efficient representation and approximation of model predictive control laws via deep learning](https://mlsanity.com/paper/1806.10644/) (2018)
- [Physics-Informed Neural Networks for AC Optimal Power Flow](https://mlsanity.com/paper/2110.02672/) (2021)
- [Learning for Constrained Optimization: Identifying Optimal Active Constraint Sets](https://mlsanity.com/paper/1802.09639/) (2018)
- [End-to-End Feasible Optimization Proxies for Large-Scale Economic Dispatch](https://mlsanity.com/paper/2304.11726/) (2023)
- [Learning for DC-OPF: Classifying active sets using neural nets](https://mlsanity.com/paper/1902.05607/) (2019)
- [Physics-Informed Neural Networks with Hard Linear Equality Constraints](https://mlsanity.com/paper/2402.07251/) (2024)
- [A Deep Reinforcement Learning Framework For Column Generation](https://mlsanity.com/paper/2206.02568/) (2022)
- [Statistical Learning For DC Optimal Power Flow](https://mlsanity.com/paper/1801.07809/) (2018)
- [Enhancing Column Generation by a Machine-Learning-Based Pricing Heuristic for Graph Coloring](https://mlsanity.com/paper/2112.04906/) (2021)
- [Partially-Supervised Neural Network Model For Quadratic Multiparametric Programming](https://mlsanity.com/paper/2506.05567/) (2025)

## Abstract (arXiv)

Deep neural networks (DNNs) have been used to model complex optimization problems in many applications, yet have difficulty guaranteeing solution optimality and feasibility, despite training on large datasets. Training a NN as a surrogate optimization solver amounts to estimating a global solution function that maps varying problem input parameters to the corresponding optimal solutions. Work in multiparametric programming (mp) has shown that solutions to quadratic programs (QP) are piece-wise linear functions of the parameters, and researchers have suggested leveraging this property to model mp-QP using NN with ReLU activation functions, which also exhibit piecewise linear behaviour. This paper proposes a NN modeling approach and learning algorithm that discovers the exact closed-form solution to QP with linear constraints, by analytically deriving NN model parameters directly from the problem coefficients without training. Whereas generic DNN cannot guarantee accuracy outside the training distribution, the closed-form NN model produces exact solutions for every discovered critical region of the solution function. To evaluate the closed-form NN model, it was applied to DC optimal power flow problems in electricity management. In terms of Karush-Kuhn-Tucker (KKT) optimality and feasibility of solutions, it outperformed a classically trained DNN and was competitive with, or outperformed, a commercial analytic solver (Gurobi) at far less computational cost. For a long-range energy planning problem, it was able to produce optimal and feasible solutions for millions of input parameters within seconds.