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

Abstract

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. I. INTRODUCTIONRecently, there have been increasing interests in developing neural network schemes, including those based on deep neural networks (DNN), to solve constrained optimization problems in various problem domains, especially those needed to be solved repeatedly in real-time. The idea behind these machine learning approaches is to leverage the universal approximation capability of DNNs [1]- [3] to learn the mapping between the input parameters to the

Model-Based Deep Learning: On the Intersection of Deep Learning and Optimization

“…The latter approach, of applying a model-based optimizer to features extracted by a DNN as in Example 19, can also be used to enforce decisions made by a DNN to comply to some underlying physical requirements, see, e.g., (33).…”

Model-Based Deep Learning: On the Intersection of Deep Learning and Optimization

“…Here, (11) becomes (25) and the tuning is done via end-to-end training as in (24). The latter approach, of applying a model-based optimizer to features extracted by a DNN as in Example 19, can also be used to enforce decisions made by a DNN to comply to some underlying physical requirements, see, e.g., [34].…”

Topology-aware Graph Neural Networks for Learning Feasible and Adaptive ac-OPF Solutions

“…Regularization can greatly enhance the performance of NN models by mitigating data overfitting and improving the training speed. Simlarly for OPF learning, regularization has been introduced to e.g., improve the constraint satisfaction [9], or approach the first-order optimality [6], [10]. Nonetheless, the flow constraint in (1e) is by and large the most critical feasibility condition of ac-OPF, motivating us to develop a new ac-feasibility regularization (FR) approach.…”

“…network-wide line limit constraints. For example, a feasible domain technique was developed in [5], [9], while KKT conditions for ac/dc-OPF were used for training regularization [6], [10]. While the first approach could affect the OPF solution optimality, the second one tends to add a large number of regularization terms, all of which require the design of weight coefficients (hyper-parameters).…”