  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.
