dOPT：通过几何约简对锥优化进行微分

中文摘要 AI 辅助

优化层能够将结构化约束和决策问题纳入学习系统。训练此类系统需要对嵌入的优化问题进行微分，这对于一般的锥规划问题可能具有挑战性。我们提出dOPT，一个与求解器无关的框架，它不是对完整的锥形式进行微分，而是在计算出的原始-对偶解处将其约简为等式约束的二次规划，该二次规划保留了参考解及其一阶敏感性。该约简捕获了与微分相关的局部一阶和二阶锥几何，并且在奇异配置下仍然定义良好。计算解的导数则只需要一次对称线性求解，与正向求解器无关。我们为凸非线性规划、二次规划、二阶锥规划和半定规划推导了显式约简。数值实验验证了计算梯度的正确性，并展示了有利的反向传播可扩展性，随着问题规模的增大，与现有可微锥优化方法相比有显著加速。

英文摘要

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.

发表机构

• University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）

机构由 AI 辅助整理，请以论文原文为准。