Ensuring DNN Solution Feasibility for Optimization Problems with Convex Constraints and Its Application to DC Optimal Power Flow Problems

Ensuring solution feasibility is a key challenge in developing Deep Neural Network (DNN) schemes for solving constrained optimization problems, due to inherent DNN prediction errors. In this paper, we propose a "preventive learning'" framework to systematically guarantee DNN solution feasibility for problems with convex constraints and general objective functions. We first apply a predict-and-reconstruct design to not only guarantee equality constraints but also exploit them to reduce the number of variables to be predicted by DNN. Then, as a key methodological contribution, we systematically calibrate inequality constraints used in DNN training, thereby anticipating prediction errors and ensuring the resulting solutions remain feasible. We characterize the calibration magnitudes and the DNN size sufficient for ensuring universal feasibility. We propose a new Adversary-Sample Aware training algorithm to improve DNN's optimality performance without sacrificing feasibility guarantee. Overall, the framework provides two DNNs. The first one from characterizing the sufficient DNN size can guarantee universal feasibility while the other from the proposed training algorithm further improves optimality and maintains DNN's universal feasibility simultaneously. We apply the preventive learning framework to develop DeepOPF+ for solving the essential DC optimal power flow problem in grid operation. It improves over existing DNN-based schemes in ensuring feasibility and attaining consistent desirable speedup performance in both light-load and heavy-load regimes. Simulation results over IEEE Case-30/118/300 test cases show that DeepOPF+ generates $100\%$ feasible solutions with $<$0.5% optimality loss and up to two orders of magnitude computational speedup, as compared to a state-of-the-art iterative solver.