dOPT: Differentiating Conic Optimization via Geometric Reduction
Abstract
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.
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BibTeX
@misc{yang2026dopt,
title = {{dOPT: Differentiating Conic Optimization via Geometric Reduction}},
author = {Fengyu Yang and Connor W. Magoon and Tyler Watts and Shahar Z. Kovalsky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33828},
eprint = {2609.33828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33828},
}
RIS
TY - UNPB
TI - dOPT: Differentiating Conic Optimization via Geometric Reduction
AU - Fengyu Yang
AU - Connor W. Magoon
AU - Tyler Watts
AU - Shahar Z. Kovalsky
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33828
UR - https://arxiv.org/abs/2609.33828
AB - 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.
ER -