Is there any good convex optimization problem based upperbound for the following nonlinear 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 upperbound \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.
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$\begingroup$ It is a bit difficult because the least convex (down) majorant of $\log\frac{1+x+y}{1+y}$ in the first quadrant is exactly $\log(1+x)$, and you are essentially asking about the least convex majorant of the sum of such functions in the domain $x\ge0$, $y\ge Bx$ with general matrix $B$ with positive entries so you'll have to incorporate your $\beta$ into the objective function in some essential way to get anything useful. Well, you, probably, know that already :) $\endgroup$– fedjaSep 16, 2020 at 22:47
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