Consider this following scenario. Suppose I have $N$ cents, and I want to dispatch these money to $n$ people, each got $x_i$ cents. In order to simplify this problem, we assume the cents are "divisable", which means $x_i$ can be any non-negtive real number($x_i>=0, x_i\in R$)
For any one got the money, will have some kind of "happiness index" which is directly related to the money he just got, we defined it as $f_i(x_i)$, which means different people may have different "happiness functions"
Then, here is the problem:
How can I properly divide theses money in order to got the maximum sum of "happiness index"?
$$ \max_{i=1,2,..n} \sum_{i=1}^n f_i(x_i) \quad s.t \quad \sum_{i=1}^n x_{i} = N,\quad x_i \in [0,N] $$
Here $f_i(x_i)$ is a bunch of functions, which may have some same kind of forms(but with respective parameter, for example $f_i(x_i) = N(\mu_i,\sigma_i)$ or so.
Is there any mature optimization algorithms for this specific kinds of problem(rather than some more general ones)?
Thank you for any suggestions