Optimization with Region-Reduced ReLU Neural Networks

Abstract

Optimization of mathematical models involving integer decisions and neural networks with ReLU activation (ReLU ANNs) is a challenging task. Nevertheless, such models are an enabling technology in many application domains. A prominent example is superstructure optimization in chemical engineering, where ReLU ANNs are frequently employed as surrogate models for complex nonlinear processes. We survey recent developments in this area. We argue that in addition to network size and training options of the ANNs, the ReLU activation geometry and the number of linear regions on the domain of interest have a strong impact on computational optimization performance. While standard model compression approaches such as structured pruning reduce network size, they do not explicitly address geometric considerations. Therefore, we propose a novel \emph{region-reduced} model compression approach that combines the stabilization of unstable neurons and the merging of redundant neurons to reduce the number of linear regions while maintaining predictive accuracy through error compensation. We evaluate our method against standard compression approaches on multiple optimization use cases. First, the two-dimensional peaks function for which we can visualize the activation geometry. Second, on optimization over individual surrogate ReLU ANNs for three chemical processes, and third, on a hybrid superstructure optimization problem that involves the three ReLU ANNs, additional process submodels, and binary variables. The results for the superstructure problem demonstrate the large potential of region reduction with a decrease of 40\% to 50\% in computational time, yielding solutions closer than 1\% to the reference at negligible effort of obtaining the compressed model.