This paper presents Dual Lagrangian Learning (DLL), a principled learning methodology for dual conic optimization proxies. DLL leverages conic duality and the representation power of ML models to provide high-duality, dual-feasible solutions, and therefore valid Lagrangian dual bounds, for linear and nonlinear conic optimization problems. The paper introduces a systematic dual completion procedure, differentiable conic projection layers, and a self-supervised learning framework based on Lagrangian duality. It also provides closed-form dual completion formulae for broad classes of conic problems, which eliminate the need for costly implicit layers. The effectiveness of DLL is demonstrated on linear and nonlinear conic optimization problems. The proposed methodology significantly outperforms a state-of-the-art learning-based method, and achieves 1000x speedups over commercial interior-point solvers with optimality gaps under 0.5\% on average.

翻译：本文提出了对偶拉格朗日学习（Dual Lagrangian Learning, DLL），一种用于对偶锥优化代理的、基于原理的学习方法。DLL 利用锥对偶性和机器学习模型的表示能力，为线性和非线性锥优化问题提供高对偶性、对偶可行的解，从而获得有效的拉格朗日对偶界。本文介绍了一种系统性的对偶补全过程、可微的锥投影层，以及一个基于拉格朗日对偶的自监督学习框架。同时，为广泛的锥优化问题类别提供了闭式对偶补全公式，从而消除了对计算成本高昂的隐式层的需求。DLL 的有效性在线性和非线性锥优化问题上得到了验证。所提出的方法显著优于当前最先进的基于学习的方法，并且相比商业内点法求解器，在平均最优间隙低于 0.5% 的情况下，实现了 1000 倍的加速。