An analysis of optimization problems involving ReLU neural networks

Abstract Solving mixed-integer optimization problems with embedded neural networks with ReLU activation functions is challenging. Big-M coefficients that arise in relaxing binary decisions related to these functions grow exponentially with the number of layers. We survey and propose different approaches to analyze and improve the run time behavior of mixed-integer programming solvers in this context. Among them are clipped variants of the activation function and regularization techniques applied during training as well as optimization-based bound tightening and a novel scaling for given ReLU networks. We numerically compare these approaches for three benchmark problems from the literature and discuss cost-benefit tradeoffs of the post-processing methods. We use the number of linear regions, the percentage of stable neurons, and overall computational effort as indicators. As a major takeaway we observe and quantify a trade-off between the often desired redundancy of neural network models versus the computational costs for solving related optimization problems.