When the first derivative of f function is equal to 0 for \tau_0, we can have
why do we have it?
I don’t know what your question is.
If you have trouble understanding the Lambert W function, you should seek help at a more appropriate place, such as https://math.stackexchange.com/ .
If you want to know how to implement the Lambert W function in CVX, there are several posts listed at http://ask.cvxr.com/search?q=lambert%20W .