  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.
