[Math] Maximize sum of logs subject to constraints

Question

I have the optimization problem

$$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{i=1}^n \log(c_i + x_i)\\ \text{subject to} & \displaystyle\sum_{i=1}^n x_i = 1\\ & x_i\ge0\end{array}$$

where $c_i>0$. How could I solve it?

Answer 1

Maximizing your function is the same as maximizing its exponential, that is: $$\prod_{i=1}^{n}\left(c_i+x_i\right)\stackrel{AM-GM}{\leq}\left(\frac{1}{n}\sum_{i=1}^{n}(c_i+x_i)\right)^{n} =\frac{1}{n^n}\left(1+\sum_{i=1}^{n}c_i\right)^n.$$ Can you take it from here?