my the optimization problem as follows:
$f\left(x \right) = \frac{\log_2\left( 1+ x\right)}{1+\exp\left(x\right) + 0.1 \log_2\left( 1+ x\right)} $
The presented problem is the fractional programming problem so I can use e.g. the Dinkelbach method but in order to do it the numerator has to be nonnegative and concave while the denominator has to be positive and convex or if the numerator is affine, the denominator does not have to be restricted in sign.
In my case, the numerator is concave while the denominator is a nonconvex function. So, can I add the constraint which removes a convex function from the numerator and looks like this:
$f\left(x \right) = \frac{y}{1+\exp\left(x\right) + 0.1 y}$
s.t.
$\log_2\left( 1+ x\right) \geq y$
After this transformation the objective function is convex and the constraint is convex, thus the optimization problem is convex. Is it correct?
The function
$$ f(x,y) = \frac{y}{1+e^x+0.1y} $$
has not compact support as can be depicted from the region (in light blue) obeying
$$ f(x,y) > 0 $$