# Standard cumulative distribution function with optimization model variable

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?

• What do you men by "characteristic function of the standard deviation", care to explain? – kjetil b halvorsen Oct 7 '19 at 21:59
• Is it me or English lacks this definition? "Loi normale centrée réduite" (in French) is what I mean. The "default" standard law, when the average is 0 and the standard error is 1. – V. Brunelle Oct 7 '19 at 22:17
• I do not speak french, so let us wait for some french speaker to show up ... – kjetil b halvorsen Oct 7 '19 at 22:30
• Is $\Phi(x)$ the CDF? As in $\Phi_X(x) = P(X\le x)$ for the standard Normal distribution for $X\sim N(\mu =0, \sigma = 1)$, $\text{E}[X]=\mu$ and $\sqrt{\text{Var}(X)}=\sigma$? – SecretAgentMan Oct 8 '19 at 0:11
• @SecretAgentMan Yes it is. µ=0 and sigma=1. – V. Brunelle Oct 8 '19 at 0:32

For strictly increasing CDFs, you can invert: $$x \le \Phi^{-1}(b)$$