Closed Form Solution for Minimization involving Standard Normal CDF and PDF Could someone please advice and provide detailed steps regarding any possible closed form solutions or other suggestions regarding solving a minimization problem of the type shown below? Here, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively. $\sigma$ is the standard deviation and $\theta$ is a parameter. We need to find the solution by expressing $x$ in terms of $y$ and the other parameters.

\begin{eqnarray*}
=\underset{\left\{ x\right\} }{\min}\left\{ x\sigma\left[\frac{\theta x}{\sigma}+\frac{\phi\left(\frac{\theta x}{\sigma}\right)}{\Phi\left(\frac{\theta x}{\sigma}\right)}\right]+\left(y-x\right)\sigma\left[\frac{\theta\left(y-x\right)}{\sigma}+\frac{\phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}{\Phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}\right]\right\} 
\end{eqnarray*}

First Order Conditions (FOC) would give the below. Please point out ways to simplify or alternative approaches as well.

\begin{eqnarray*}
x\left[\frac{4\theta}{\sigma}+\theta\left\{ \frac{\phi'\left(\frac{\theta x}{\sigma}\right)}{\Phi\left(\frac{\theta x}{\sigma}\right)}-\phi'\left(\frac{\theta x}{\sigma}\right)\left[\frac{\phi\left(\frac{\theta x}{\sigma}\right)}{\Phi\left(\frac{\theta x}{\sigma}\right)}\right]^{2}\right\} \right.\\
+\left.\theta\left\{ \frac{\phi'\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}{\Phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}-\phi'\left(\frac{\theta\left(y-x\right)}{\sigma}\right)\left[\frac{\phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}{\Phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}\right]^{2}\right\} \right] &amp; =\\
\left(\theta y\right)\left\{ \frac{2}{\sigma}+\frac{\phi'\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}{\Phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}-\phi'\left(\frac{\theta\left(y-x\right)}{\sigma}\right)\left[\frac{\phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}{\Phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}\right]^{2}\right\} \\
+\frac{\sigma\phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}{\Phi\left(\frac{\theta\left(y-x\right)}{\sigma}\right)}-\frac{\sigma\phi\left(\frac{\theta x}{\sigma}\right)}{\Phi\left(\frac{\theta x}{\sigma}\right)}
\end{eqnarray*}
Looks like a 1-variable problem.

For analytical approach , how about taking one derivative in $x$ and setting it equal to zero and see what you can get.

For a numerical approach, since you have the derivative (and even two if you like), you can call a minimization algo, or even rootfind the derivative with Newton's method or Brent or Secant method. This will, unfortunately, only work, for fixed values of $\theta, y$.
Looks like a 1-variable problem.

For analytical approach , how about taking one derivative in $x$ and setting it equal to zero and see what you can get.

For a numerical approach, since you have the derivative (and even two if you like), you can call a minimization algo, or even rootfind the derivative with Newton's method or Brent or Secant method. This will, unfortunately, only work, for fixed values of $\theta, y$.