I have been researching and the proximal operator seems to be defined as:
I have found it stated several times that this is always a convex problem with a unique minimizer. I have not, however, been able to find a proof of this statement. So, why does the proximal operator have a unique solution?
In addition, this article (link credit Mark L. Stone, see comments) hints that $f(x)$ must be convex for the proximal operator to be a convex problem. If this condition is required by the proof, why would someone use proximal optimization methods over simple gradient descent?