How to implement new convex support for log function?

1

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?

2

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.