# dOPT: Differentiating Conic Optimization via Geometric Reduction

arXiv: https://arxiv.org/abs/2609.33828 · HTML: https://mlsanity.com/paper/2609.33828/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 37 of its 57 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 30 of 37)

- [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)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [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)
- [Differentiating Through a Cone Program](https://mlsanity.com/paper/1904.09043/) (2019)
- [Nonsmooth Implicit Differentiation for Machine Learning and Optimization](https://mlsanity.com/paper/2106.04350/) (2021)
- [On the Differentiability of the Solution to Convex Optimization Problems](https://mlsanity.com/paper/1804.05098/) (2018)
- [Alternating Differentiation for Optimization Layers](https://mlsanity.com/paper/2210.01802/) (2022)
- [Efficient differentiable quadratic programming layers: an ADMM approach](https://mlsanity.com/paper/2112.07464/) (2021)
- [Flexible Differentiable Optimization via Model Transformations](https://mlsanity.com/paper/2206.06135/) (2022)
- [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)
- [SDPRLayers: Certifiable Backpropagation Through Polynomial Optimization Problems in Robotics](https://mlsanity.com/paper/2405.19309/) (2024)
- [LPGD: A General Framework for Backpropagation through Embedded Optimization Layers](https://mlsanity.com/paper/2407.05920/) (2024)
- [SCQPTH: an efficient differentiable splitting method for convex quadratic programming](https://mlsanity.com/paper/2308.08232/) (2023)
- [Differentiating Through a Quadratic Cone Program](https://mlsanity.com/paper/2508.17522/) (2025)
- [Decision-Focused Learning via Tangent-Space Projection of Prediction Error](https://mlsanity.com/paper/2605.01361/) (2026)
- [Code generation for solving and differentiating through convex optimization problems](https://mlsanity.com/paper/2504.14099/) (2025)
- [A General and Streamlined Differentiable Optimization Framework](https://mlsanity.com/paper/2510.25986/) (2025)
- [A Fully First-Order Layer for Differentiable Optimization](https://mlsanity.com/paper/2512.02494/) (2025)
- [Sensitivity analysis for parametric nonlinear programming: A tutorial](https://mlsanity.com/paper/2504.15851/) (2025)
- [A Penalty Approach for Differentiation Through Black-Box Quadratic Programming Solvers](https://mlsanity.com/paper/2602.14154/) (2026)
- [Scalable Deep Unfolding of Conic Optimizers](https://mlsanity.com/paper/2606.13825/) (2026)
- [Computation of graphical derivatives of normal cone maps to a class of conic constraint sets](https://mlsanity.com/paper/1709.05592/) (2017)
- [Parabolic Regularity in Geometric Variational Analysis](https://mlsanity.com/paper/1909.00241/) (2019)
- [A Strict Complementarity Approach to Error Bound and Sensitivity of Solution of Conic Programs](https://mlsanity.com/paper/2012.00183/) (2020)
- [Weak notions of nondegeneracy in nonlinear semidefinite programming](https://mlsanity.com/paper/2012.14810/) (2020)
- [On the weak second-order optimality condition for nonlinear semidefinite and second-order cone programming](https://mlsanity.com/paper/2208.03076/) (2022)

## Abstract (arXiv)

Optimization layers enable the incorporation of structured constraints and decision problems into learning systems. Training such systems requires differentiating through the embedded optimization problem, which can be challenging for general conic programs. We introduce dOPT, a solver-agnostic framework that, rather than differentiating the full conic formulation, reduces it at a computed primal-dual solution to an equality-constrained quadratic program that preserves the reference solution and its first-order sensitivity. The reduction captures the local first- and second-order conic geometry relevant to differentiation and remains well defined at singular configurations. Computing solution derivatives then requires a single symmetric linear solve, independently of the forward solver. We derive explicit reductions for convex NLPs, QPs, SOCPs, and SDPs. Numerical experiments validate the computed gradients and show favorable backward-pass scalability, with substantial speedups over existing differentiable conic optimization methods as problem size increases.