Applications of the general Hilbert projection theorem

Question

The Hilbert projection theorem asserts that if $C$ is a closed convex subset of a Hilbert space $H$ and $x \in H$, there exists a unique $y \in C$ minimizing the distance to $x$. This theorem applied to $C$ which is a linear subspace can be used to derive a large portion of the elementary Hilbert space theory, e.g. all closed subspaces are complemented or Riesz representation theorem.

Could someone provide applications of the theorem in which a general $C$ is needed? (They do not have to be limited to abstract Hilbert space theory; examples involving concrete optimization problems from natural sciences etc. are welcome).

Answer 1

• You can derive various results in Convex Analysis in Hilbert space by using the projection theorem using general convex sets. The epigraph of a convex function (a nonlinear convex set) plays a major role. For more on this, see the book Convex Analysis and Monotone Operator Theory in Hilbert Spaces by Bauschke and Combettes.
• One standard problem in Convex Optimization is to find a minimizer $x$ of a function $f$ minimized over a nonempty closed convex set $C$. If you can compute the projection operator $P_C$, then this opens the door to subgradient or projected gradient algorithms. Popular choices are nonnegativity constraints or box constraints. For more on this, see any modern Convex Optimization books; for instance the one mentioned above or First-Order Methods in Optimization by Beck.