We all know that expressions in mathematical optimization models can't contain "black boxes" around a decision variable since everything has to be written using mathematical expressions. For example, "yes/no" decisions can't be written as "if this then that" expressions in a model, but can be written using big-$M$ constraints with a binary variable.
What is the recommended way of writing a model that must compute the value of a cumulative distribution function from a decision variable?
For example, let's suppose we have a model with the following constraint:
$$\Phi_X(x)\le b$$
where $\Phi_X(x)$ is the cumulative distribution function of the standard Normal random variable $X\sim N(\mu = 0,\sigma =1)$, $x$ is a decision variable, and $b$ is a parameter of the model.
Is there any way of converting this kind of constraint into a valid mathematical formulation? What is the approach suggested by experts? Does that kind of constraint absolutely require the use of special tools that can't be found in regular optimization solvers?
