What is the difference between `exponential` and `exp_cone`?

In many answers, e.g., How to express “x*(2^(1/x)-1)” in CVX
, exponential(1) is used.

What is the difference between exponential and exp_cone?

I have tried to find exponential in CVX manual, but I cannot find it.

There are only exp_cone, defined as

I think to minimize x*(2^{1/x}-1), I have to use the right-most code as follows.

image1063×379 9.07 KB

However, in How to express “x*(2^(1/x)-1)” in CVX
, Mark_L_Stone recommended to use the following code: {1,x,z} == exponential(1)

Please let me know what the difference is between exponential and exp_cone?
Also, why he recommended to use == instead of <in>?

Thank you.

The CVX Users’ Guide erroneously shows exp_cone. The actual implemented syntax uses exponential, as used in my answers.

I don’t believe
<in> exp_cone
works, because CVX does not recognize <in>, at last when I just tried it in CVX 2.2

Here, does == imply “in”? That is, is the phrase {1,x,z} == exponential(1) representing that a tuple (1,x,z) is in a set exp_cone?

The syntax {1,x,z} == exponential(1) works. i don’t know whether any other syntax works.

I didn’t design or develop CVX. I’m just telling you what works. I suggest you use that.