Method Alternating Projections Smoothness

Question

Suppose I have $n$ closed convex sets, $C_0, C_1, \dots, C_{n-1}$ in $\mathbb{R}^d$ such that $\cap_{i=0}^{n-1} C_i \neq \emptyset$. The method of alternating projections defines a sequence in the following way: $x_{k+1} = P_{C_{[k]}}(x_k)$ where $P_C$ is the projection onto $C$, $[k]=k \text{ mod } n$, and $x_0$ is an arbitrary point in $\mathbb{R}^d$. It is well known that the sequence $\{x_k\}$ converges to a point in $\cap_{i=0}^{n-1} C_i$. Now, define a function $f:\mathbb{R}^d\rightarrow \cap_{i=0}^{n-1} C_i$ such that $f(x_0)$ is the point to which the previously defined sequence converges, given the starting point $x_0$. I would like to know if $f$ is continuous for any choice of $C_i$'s, and if so, if it is also differentiable. Thank you!

Answer 1

1. Since each projection is nonexpansive, your mapping $\;f$ is actually nonexpansive, i.e., Lipschitz continuous with constant $1$.

2. $f$ is differentiable when each set is an affine subspace because then $f$ is affine and hence differentiable.

3. In general, $f$ is not differentiable. Consider in $\mathbb{R}$ the case when $n=1$ and $C_0=\left(-\infty,0\right]$. Then $f=P_{C_0}$ is not differentiable.