DC3: A learning method for optimization with hard constraints

Abstract

Large optimization problems with hard constraints arise in many settings, yet classical solvers are often prohibitively slow, motivating the use of deep networks as cheap "approximate solvers." Unfortunately, naive deep learning approaches typically cannot enforce the hard constraints of such problems, leading to infeasible solutions. In this work, we present Deep Constraint Completion and Correction (DC3), an algorithm to address this challenge. Specifically, this method enforces feasibility via a differentiable procedure, which implicitly completes partial solutions to satisfy equality constraints and unrolls gradient-based corrections to satisfy inequality constraints. We demonstrate the effectiveness of DC3 in both synthetic optimization tasks and the real-world setting of AC optimal power flow, where hard constraints encode the physics of the electrical grid. In both cases, DC3 achieves near-optimal objective values while preserving feasibility.

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

“…For example, existing works belong to the "learn to optimize" field, using RNN to mimic the gradient descent-wise iteration and achieve faster convergence speed empirically [43], [44]. Other works like [6], [7], [13] directly used the DNN model to predict the final solution (regarded as end-to-end method), which can further reduce the computing time compared to the iteration-based approaches. These approaches, in general, can have better speedup performance compared with the hybrid approaches.…”

“…While the projection based post-processing step can retrieve a feasible solution in the face of infeasibility, the scheme turns to be computationally expensive and inefficient. Several techniques that incorporate problem constraints during training to improve DNN feasibility are proposed [6], [7]. These approaches enforce equality constraints by predicting a set of variables and recovering the remaining ones from such equality equations.…”

“…1. The framework first adopts the predict-and-reconstruct idea [6], [7], [13] to guarantee the equality constraints satisfaction by predicting a subset of N − p variables in P and recovering the remaining p ones by solving the linear equations in (2). This way, we not only guarantee equality constraints but also exploit them to reduce the number of variables to be predicted by DNN.…”

“…is the applied ReLU activation functions 7 in the hidden layers and the output layer. Here the max(x, 0) means taking element-wise max operation over the included vector.…”

“…( 9) and ( 10), k = 1, ..., N hid represent the correlation between the input θ of the DNN to its predicted variables x via the formulated DNN constraints. ( 12) denotes the relative violation on each constraint considering the limits calibration, where êj = (1 ej ≥0 (1−∆)+1 ej <0 (1+∆))•e j denotes the constraint limit after calibration and ∆ represents the determined calibration rate via ( 6)- (7). Here recall that for the class of OPLC, the obtained ∆ is no greater than the maximal one 9 with which the target input parameter region is still guaranteed to be preserved.…”