I am trying to find a continuous function x(t) defined over non-negative 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 differentiable 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.