  Deep neural networks (DNNs) are widely studied in various applications. A DNN
consists of layers of neurons that compute affine combinations, apply nonlinear
operations, and produce corresponding activations. The rectified linear unit
(ReLU) is a typical nonlinear operator, outputting the max of its input and
zero. In scenarios like max pooling, where multiple input values are involved,
a fixed-parameter DNN can be modeled as a mixed-integer program (MIP). This
formulation, with continuous variables representing unit outputs and binary
variables for ReLU activation, finds applications across diverse domains. This
study explores the formulation of trained ReLU neurons as MIP and applies MIP
models for training neural networks (NNs). Specifically, it investigates
interactions between MIP techniques and various NN architectures, including
binary DNNs (employing step activation functions) and binarized DNNs (with
weights and activations limited to $-1,0,+1$). The research focuses on training
and evaluating proposed approaches through experiments on handwritten digit
classification models. The comparative study assesses the performance of
trained ReLU NNs, shedding light on the effectiveness of MIP formulations in
enhancing training processes for NNs.
