# BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning arXiv: https://arxiv.org/abs/2411.19285 ยท HTML: https://mlsanity.com/paper/2411.19285/ ## Key facts - Cited by 10 papers in the map (data through 14 Oct 2026). - Builds on 28 of its 43 references that are in the map. - First posted on arXiv 28 Nov 2024. ## Builds on (most cited first, 28 of 28) - [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018) - [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016) - [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017) - [Learning Combinatorial Optimization Algorithms over Graphs](https://mlsanity.com/paper/1704.01665/) (2017) - [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019) - [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017) - [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018) - [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021) - [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021) - [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019) - [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017) - [Learning the Travelling Salesperson Problem Requires Rethinking Generalization](https://mlsanity.com/paper/2006.07054/) (2020) - [Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions](https://mlsanity.com/paper/2106.01798/) (2021) - [MIPaaL: Mixed Integer Program as a Layer](https://mlsanity.com/paper/1907.05912/) (2019) - [Smart Predict-and-Optimize for Hard Combinatorial Optimization Problems](https://mlsanity.com/paper/1911.10092/) (2019) - [Qlib: An AI-oriented Quantitative Investment Platform](https://mlsanity.com/paper/2009.11189/) (2020) - [Deep Declarative Networks: A New Hope](https://mlsanity.com/paper/1909.04866/) (2019) - [Combinatorial Optimization by Graph Pointer Networks and Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1911.04936/) (2019) - [Differentiating Through a Cone Program](https://mlsanity.com/paper/1904.09043/) (2019) - [Improved SVRG for Non-Strongly-Convex or Sum-of-Non-Convex Objectives](https://mlsanity.com/paper/1506.01972/) (2015) - [Automatically Learning Compact Quality-aware Surrogates for Optimization Problems](https://mlsanity.com/paper/2006.10815/) (2020) - [Alternating Differentiation for Optimization Layers](https://mlsanity.com/paper/2210.01802/) (2022) - [Operator splitting for a homogeneous embedding of the linear complementarity problem](https://mlsanity.com/paper/2004.02177/) (2020) - [SurCo: Learning Linear Surrogates For Combinatorial Nonlinear Optimization Problems](https://mlsanity.com/paper/2210.12547/) (2022) - [Efficient differentiable quadratic programming layers: an ADMM approach](https://mlsanity.com/paper/2112.07464/) (2021) - [End-to-End Stochastic Optimization with Energy-Based Model](https://mlsanity.com/paper/2211.13837/) (2022) - [Landscape Surrogate: Learning Decision Losses for Mathematical Optimization Under Partial Information](https://mlsanity.com/paper/2307.08964/) (2023) - [Decision-Focused Learning without Differentiable Optimization: Learning Locally Optimized Decision Losses](https://mlsanity.com/paper/2203.16067/) (2022) ## Built on by (most cited first, 10 of 10) - [A Fully First-Order Layer for Differentiable Optimization](https://mlsanity.com/paper/2512.02494/) (2025) - [A Penalty Approach for Differentiation Through Black-Box Quadratic Programming Solvers](https://mlsanity.com/paper/2602.14154/) (2026) - [Learning to Optimize by Differentiable Programming](https://mlsanity.com/paper/2601.16510/) (2026) - [SOC-ICNN: From Polyhedral to Conic Geometry for Learning Convex Surrogate Functions](https://mlsanity.com/paper/2604.22355/) (2026) - [BOCO: Bayesian Online Contextual Optimization for Decision-Focused Online Learning](https://mlsanity.com/paper/2511.20413/) (2025) - [Integrated Prediction and Multi-period Portfolio Optimization](https://mlsanity.com/paper/2512.11273/) (2025) - [Iskra: A System for Inverse Geometry Processing](https://mlsanity.com/paper/2602.12105/) (2026) - [Diff2SP: Diffusion Models for Correlated Scenario Generation in Stochastic Programming](https://mlsanity.com/paper/2606.05649/) (2026) - [Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics](https://mlsanity.com/paper/2606.16564/) (2026) - [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026) ## Abstract (arXiv) Data-driven decision-making processes increasingly utilize end-to-end learnable deep neural networks to render final decisions. Sometimes, the output of the forward functions in certain layers is determined by the solutions to mathematical optimization problems, leading to the emergence of differentiable optimization layers that permit gradient back-propagation. However, real-world scenarios often involve large-scale datasets and numerous constraints, presenting significant challenges. Current methods for differentiating optimization problems typically rely on implicit differentiation, which necessitates costly computations on the Jacobian matrices, resulting in low efficiency. In this paper, we introduce BPQP, a differentiable convex optimization framework designed for efficient end-to-end learning. To enhance efficiency, we reformulate the backward pass as a simplified and decoupled quadratic programming problem by leveraging the structural properties of the KKT matrix. This reformulation enables the use of first-order optimization algorithms in calculating the backward pass gradients, allowing our framework to potentially utilize any state-of-the-art solver. As solver technologies evolve, BPQP can continuously adapt and improve its efficiency. Extensive experiments on both simulated and real-world datasets demonstrate that BPQP achieves a significant improvement in efficiency--typically an order of magnitude faster in overall execution time compared to other differentiable optimization layers. Our results not only highlight the efficiency gains of BPQP but also underscore its superiority over differentiable optimization layer baselines.