Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$.
Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ has at least one fixed point in $X$.)
Under reasonable assumptions on $f$, which fixed point iteration converge to the fixed point of $f$ that is (among) the closest to $Y$?
(Partial/Approximate Answers and/or Hints are also OK!)