I am having a problem of maximizing a sum of sigmoid functions over different time instants with some constraints.
Considering the standard sigmoid function $f(x)=\frac{1}{1+e^{-\alpha x}}$ and it's derivative $f'(x)=f(x)[1-f(x)]$
In my case, it is slightly different, the sigmoid function at time instant $n$ is defined as $f_n(x)=\frac{1}{1+e^{-\alpha(\frac{x}{y_n}-z)}}$, where z>0 and $\alpha>0$ are constants and $y_n>0$ is defined for every time instant $n$.
I need to find $x$ that maximize the following:
$\sum_{n=1}^N \frac{1}{1+e^{-\alpha(\frac{x}{y_n}-z)}}\:\: -c x \:\:$ such that $\:\:x \leq X$, $\:\:\:(1)$
where $c>0$ is some constant represents a cost value, $X$ is another constant represents the maximum value of $x$ and $N$ is total number of time instants.
One of the solution I was thinking about is to use Lagrangian:
$L(x,\lambda)=\sum_{n=1}^N \frac{1}{1+e^{-\alpha(\frac{x}{y_n}-z)}}\:\: -c x \:\:- \lambda (x-X)$
where $\lambda$ is the Lagrange multiplier, since we can find the derivative of equation (1) over $x$.
I tried this method but after getting $\frac{\partial L(x,\lambda)}{\partial x}=0 \:\:$ I could't solve it.
I am not sure this type of optimization problem (1) is solvable or not. And if not is there some type of approximation/relaxation can be used to solve it.
I don't have much experience in optimization problem. So, I was hopping to get some help.
-Thanks
