Neural Nework and mathematical programming I am solving a scheduling problem where staff members cannot have overtime. For this problem, I have an integer programming model and solve it using the CPLEX solver. A simplified version of this model looks like the below:
$$\begin{align*}
\min Z &= X C^\mathsf T\\
XB^\mathsf T &\geq b\\
X &\in Z_{\geq0}\end{align*}
$$
Given the complexity of the problem, I would like to approximate the overtime component of the problem for each given solution $X$ (e.g., the number of scheduled patients). For this, I have built a simulation model, run it for several different potential values of $X$ (input values) and calculated the total overtime (output values). Now, I need a way to build a LINEAR approximation equation in $X$ and include it in the above optimization model. One way is to use regression analysis and fit an algebraic model to predict the overtime.
However, I would like to know whether I could use a neural network to derive a linear algebraic equation in $X$ and predict the overtime for different values of $X$? I have found this article, but it seems that the resulting equations are not linear due to the presence of activation functions. Note that I do not want to use the neural network to solve the problem. I just want to use the neural network to derive a linear algebraic equation in $X$ and include it as a constraint set in the above model.

We can use the ReLU activation function and derive a linear equation from a neural network. Further information is presented in this paper (doi: 10.1007/s10601-018-9285-6)
