I am trying to find a continuous function $x(t)$ defined over positive real numbers that minimizes the expression below:
$$\frac{\nu y_1 + (1-\nu)y_2}{y_0}$$
where
- $\displaystyle \nu = \int_{0}^{\infty} (1-x(t)) f'(t) dt$,
- $\displaystyle y_0 = \int_{0}^{\infty} (1-f(t))(1-x(t)) dt$,
- $\displaystyle y_1 = \int_{0}^{\infty} (1 - \frac{x(t)}{C_1 x(t) + 1}) dt$,
- $\displaystyle y_2 = C_2$,
$C_1$ and $C_2$ are positive real constants, $f(t)$ is a probability density function. $f(t)$ is differential everywhere. And I need to find a function $x(t):\mathbb{R^+}\cup \{0\}\to\mathbb{R^+}$.
I tried to see whether the Euler equation from the calculus of variations can help. However I could not find a way to progress.