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.