Why, in general, someone should be interested in finding projections onto epigraph(f), where $f:X\to\mathbb{R}$ is a given function? ($X$ Hilbert space).
I heard about that in problems related to convex optimization, but I don't understand why this issue is so important.
I hope someone could help me. Thank you in advance!
${\bf EDIT:}$ I refer to these notes: https://gubner.ece.wisc.edu/notes/ConvexityNotes.pdf
At the end of page 5, the author says that: "Since we will be interested in projections onto epi f , itis important to have conditions on f under which epi($f$) will be nonempty, closed, and convex." I don't understand why he is interested in that.