Solving equations with Standard Normal CDF and PDF (Optimization) How do we go about solving equations of this sort, where we need to find $x$ satisfying the below? Here $K$ and $\xi$ are known constants. Also, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively.
\begin{eqnarray*}
\frac{\xi\left(K-x\right)^{2}\phi\left(\xi\left\{ K-x\right\} \right)}{\Phi\left(\xi\left\{ K-x\right\} \right)}+\left(K-x\right)\left[\frac{\phi\left(\xi\left\{ K-x\right\} \right)}{\Phi\left(\xi\left\{ K-x\right\} \right)}\right]^{2} &  & =\\
\left\{ K-2x\right\} +\frac{1}{\xi}\frac{\phi\left(\xi\left\{ K-x\right\} \right)}{\Phi\left(\xi\left\{ K-x\right\} \right)}+\frac{\xi Kx\phi\left(\xi x\right)}{\Phi\left(\xi x\right)}+K\left[\frac{\phi\left(\xi x\right)}{\Phi\left(\xi x\right)}\right]^{2}
\end{eqnarray*}
This comes up during the minimization of this problem.
\begin{eqnarray*}
\underset{\left\{ x\right\} }{\min}\left[K\left\{ \xi x+\frac{\phi\left(\xi x\right)}{\Phi\left(\xi x\right)}\right\} +\left(K-x\right)\left\{ \xi\left(K-x\right)+\frac{\phi\left(\xi\left(K-x\right)\right)}{\Phi\left(\xi\left(K-x\right)\right)}\right\} \right]
\end{eqnarray*}
Please note this can be shown to be convex and there is a separate thread on this.
Convexity of Function of PDF and CDF of Standard Normal Random Variable
 A: Denote by $\lambda(x)$ the Inverse Mill's ratio
$$\lambda(x) := \frac{\phi(x)}{1 - \Phi(x)} = \frac{\phi(-x)}{\Phi(-x)}.$$
Your objective can be rephrased as
$$\min_{x} K\bigg( \xi x+\lambda(-\xi x) \bigg) +\left(K-x\right)\bigg( \xi\left(K-x\right)+\lambda(-\xi\left(K-x \right))\bigg).$$
I don't see how convexity follows from the question you linked, since that question restricts the domain to $x \geq 0$ and focuses on the right term.
But, assuming you've convinced yourself of the convexity, such a function is easy to minimize numerically. The convexity gives you that any stationary point is a global minimizer. Here's how to do the optimization in Python.
from scipy import optimize
from scipy import stats

def inv_mills(x):
    return scipy.stats.pdf(-x)/scipy.stats.cdf(-x)

#define your constants K and xi
xi = 1.0; K = 2.0;

result = optimize.minimize_scalar(lambda x: K*(xi*x + inv_mills(-xi*x)) + (K-x)*(xi*(K-x) + inv_mills(-xi*(K-x))))
print(result)

My example has the following result:
    nfev: 14
     nit: 10
 success: True
       x: 1.3308029569427138

