Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem?
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\sum_{m}\beta_{m,n}x_m})\\
\text{subject to}&\quad\mathbf{A}\mathbf{x}\leq \mathbf{p}\\
&\quad\mathbf{x}\geq\mathbf{0},
\end{align}
where $\mathbf{x}=[x_1,\ldots,x_N]^{\mathrm{T}}$ and $\beta_{m,n}$'s are positive coefficients. I know about the following upper-bound
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+x_n)\\
\text{subject to}&\quad\mathbf{A}\mathbf{x}\leq \mathbf{p}\\
&\quad\mathbf{x}\geq\mathbf{0},
\end{align}
which is very loose.