How to implement new convex support for log function?

I want to implement the support for the new convex problem of the log function, i.e., log (e^e^x-1) as following:

x=-40:0.5:6;
y=log(exp(exp(x))-1);
plot(x,y,‘r-’);

This log function is convex, though it belongs to “log-convex” except of “log convex monomial” and “log convex posynomial”.

Therefore, I want to implement the code to solve the following convex problem:

cvx_begin
variables r
minimize( t-r );
subject to
log(exp(exp®)-1)<=t+10000;
cvx_end

Obviously, this convex problem is not supported by the current cvx version.
I could implement the function “exp(exp®)-1” using the epigraph.
However, how to improve the code in basis to solve this convex
problem?

It really is not possible to handle this function I’m afraid. The underlying solvers simply cannot handle it. You would have to implement your own successive approximation approach or a polynomial approximation.