Trying to solve $\max \limits_{\mathbf{x}} \sum_{i=1}^K \log_2(1+\frac{x_i a_{ii} }{\sum_{n \ne i} x_n a_{ni} })$, s.t. $\sum_{i=1}^K x_i \le b$

Question

I am trying to solve the following optimization problem: \begin{array}{rl} \max \limits_{\mathbf{x}} & \sum_{i=1}^K \log_2(1+\frac{x_i a_{ii} }{\sum_{n \ne i} x_n a_{ni} }) \\ \mbox{subject to} & \sum_{i=1}^K x_i \le b \end{array} Here, $\mathbf{x}=[x_1,\ldots,x_K]$, and $a_{ni}$ (for all $i$ and $n$), $a_{ii}$ and $b$ are some positive constants. Note that the $x_i$ are non-negative.

How to solve this problem ?

Answer 1

If I have well understood your problem:

If you take $K=2$, your sum is $$S=\log_2(1+\frac{a_1x_1}{a_2x_2})+\log_2(1+\frac{a_2x_2}{a_1x_1})$$ Now if you suppose that $x_1=b-x_2$ with $b>x_2>0$, and you let $x_2\to 0$, it is clear that $S\to +\infty$. So in this case, the max does not exists. For $K\geq 3$, take $x_i=0$ for $i\geq 3$, and use the same procedure.