The literature has shown how to optimize and analyze the parameters of different types of neural networks using mixed integer linear programs (MILP). Building on these developments, this work presents an approach to do so for a McCulloch/Pitts and Rosenblatt neurons. As the original formulation involves a step-function, it is not differentiable, but it is possible to optimize the parameters of neurons, and their concatenation as a shallow neural network, by using a mixed integer linear program. The main contribution of this paper is to additionally enforce sparsity constraints on the weights and activations as well as on the amount of used neurons. Several experiments demonstrate that such constraints effectively prevent overfitting in neural networks, and ensure resource optimized models.