A while ago I posted this article on how to project on the epigraph of a convex function where I derived the optimality conditions and the KKT conditions. This post comes as an addendum proving a third way to project on an epigraph. Do read the previous article first because I use the same notation here.
We take the infimum which corresponds to the optimization problem that defines the projection on the epigraph and we have
where we have done the converse of an epigraphical relaxation; we define the function and notice that this is minimized at
and . This is of course only useful to the extent that is computable.