I would like to be directed to the proper literature regarding the following problem:
Given a constant sum of $x_n$ values:
$\sum_{n=1}^{N}{\left. {x_n}\right.} = C$
where $x_n ≥ 1$
Find the maximum of the following expression:
$\sum_{n=1}^{N}{\left. {a_n} \log{\left({x_n}\right) }\right.}$
where the $a_n > 0$ values are constants, and the $x_n$ values are free to change, given that their sum will remain constant.
What is the official name or category of this optimization problem? Could someone recommend a book or a website where it is explained how to solve it?
Calling $y_n = x_n - 1$ we have the equivalent problem
$$ \max \sum_{n=1}^N a_n\ln(y_n+1)\ \ \text{s. t.}\ \ \ \sum_{n=1}^N y_n = C-N $$
so the lagrangian reads
$$ L(y,\lambda) = \sum_{n=1}^N a_n\ln(y_n+1)-\lambda\left( \sum_{n=1}^N y_n - C + N\right) $$
The stationary points are the solutions for
$$ \nabla L = 0 = \left\{\begin{array}{rcl}\frac{a_n}{y_n+1}&=&\lambda\\ \sum_{n=1}^N y_n &=& C-N \end{array}\right. $$
substituting
$$ y_n = \frac{a_n}{\lambda}-1 $$
into the restriction we have
$$ \frac{1}{\lambda}\sum_{n=1}^N a_n-N = C - N $$
or
$$ \lambda = \frac 1C\sum_{n=1}^N a_n\Rightarrow y_n = \frac{a_n C}{\sum_{n=1}^N a_n}-1\Rightarrow x_n = \frac{a_n C}{\sum_{n=1}^N a_n} $$
expecting of course that $a_k, C$ are such that $\frac{a_n C}{\sum_{n=1}^N a_n}\ge 1$. Here, as $\log$ is an strict increasing function, there exist $C$ such that this positive condition is verified.