I have an optimization problem where the following constraint causes DCP Rule Error.
$$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\rho_i} g_i^2+\sigma^2}\right)$$
Assuming that $\rho_n$ and $x_n$ are among the decision variables and $B$, $\sigma$ and $g$ are constant. Is this constraint convex?
I tried to write the Python code of the optimization problem using the cvxpy library and the mosek solver, but the above clause causes a DCP Rule Error.
Rewrite the constraint as $$x_n \le f_n(\rho_1,\dots,\rho_n):=\ln\Big(B\log_2\Big(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\rho_i} g_i^2+\sigma^2}\Big)\Big).$$
The problem is then to prove the concavity of $f_n$. This seems to be a challenging problem.
We can do it for $n=2$. Indeed, in this case the problem reduces to showing that the function $f$ given by $$f(r,s):=\ln\ln\Big(1+\frac{ae^s}{b+e^r}\Big)$$ is concave for any real $a,b>0$. We have $$\partial_s^2 f(r,s) =\frac{a v (b+u)}{(b+u+a v)^2 \ln ^2\Big(1+\frac{a v}{b+u}\Big)} \Big(\frac{a v}{b+u}-\ln \Big(\frac{a v}{b+u}+1\Big)\Big) <0 $$ and $$\partial_r^2 f(r,s)\,\partial_s^2 f(r,s)-\partial_r \partial_s f(r,s)^2 =\\ \frac{a^2 u v^2 \big(b (a v+u)+b^2\big)}{(b+u) (a v+b+u)^4 \ln ^3(\frac{a v}{b+u}+1\Big)} \Big(\frac{a v}{b+u}-\ln \Big(\frac{a v}{b+u}+1\Big)\Big)>0,$$ where $u:=e^r$ and $v:=e^s$. So, $f$ is concave.
