Conventional solvers are often computationally expensive for constrained optimization, particularly in large-scale and time-critical problems. While this leads to a growing interest in using neural networks (NNs) as fast optimal solution approximators, incorporating the constraints with NNs is challenging. In this regard, we propose deep Lagrange dual with equality embedding (DeepLDE), a framework that learns to find an optimal solution without using labels. To ensure feasible solutions, we embed equality constraints into the NNs and train the NNs using the primal-dual method to impose inequality constraints. Furthermore, we prove the convergence of DeepLDE and show that the primal-dual learning method alone cannot ensure equality constraints without the help of equality embedding. Simulation results on convex, non-convex, and AC optimal power flow (AC-OPF) problems show that the proposed DeepLDE achieves the smallest optimality gap among all the NN-based approaches while always ensuring feasible solutions. Furthermore, the computation time of the proposed method is about 5 to 250 times faster than DC3 and the conventional solvers in solving constrained convex, non-convex optimization, and/or AC-OPF.

翻译：传统求解器在处理约束优化问题时通常计算成本高昂，尤其在规模庞大且对时间敏感的复杂问题中。尽管这促使学界日益关注使用神经网络作为快速最优解近似器，但如何将约束条件融入神经网络仍具挑战性。为此，我们提出带有等式嵌入的深度拉格朗日对偶（DeepLDE）框架——一种无需标签即可学习寻找最优解的方法。为确保解的可行性，我们将等式约束嵌入神经网络，并采用原始-对偶方法训练网络以施加不等式约束。进一步地，我们证明了DeepLDE的收敛性，并揭示仅凭原始-对偶学习方法无法在无等式嵌入辅助下保证等式约束的满足。在凸优化、非凸优化及交流最优潮流（AC-OPF）问题上的仿真结果表明：所提出的DeepLDE在保证解可行性的前提下，实现了所有基于神经网络方法中最小的最优性差距。此外，该方法在求解约束凸优化、非凸优化及/或AC-OPF时的计算速度较DC3及传统求解器快约5至250倍。