  We introduce a fixed point iteration process built on optimization of a
linear function over a compact domain. We prove the process always converges to
a fixed point and explore the set of fixed points in various convex sets. In
particular, we consider elliptopes and derive an algebraic characterization of
their fixed points. We show that the attractive fixed points of an elliptope
are exactly its vertices. Finally, we discuss how fixed point iteration can be
used for rounding the solution of a semidefinite programming relaxation.
