Any result on ADMM iteration being a contraction? Consider a standard ADMM problem:
minimize $f(x) + g(z)$ subject to $A x + B z = c$
The scaled ADMM algorithm is (from Boyd's paper):
$$ \begin{aligned} x^{k+1} &= \underset{x}{\operatorname{argmin}} \left( f(x) + \frac{\rho}{2} \left\| A x + B z^k - c + u^k \right\|_2^2 \right) \\\\ z^{k+1} &= \underset{z}{\operatorname{argmin}} \left( g(z) + \frac{\rho}{2} \left\| A x^{k+1} + B z - c + u^k \right\|_2^2 \right) \\\\ u^{k+1} &= u^k + A x^{k+1} + B z^{k+1} - c \end{aligned}$$
Define $\xi^k = (x^k, z^k, u^k)$. Each ADMM iteration will define a map $\Gamma$ from $\xi^k$ to $\xi^{k+1}$, that is $\xi^{k+1} = \Gamma(\xi^k)$.
Suppose we have all the sufficient conditions so that the sequence $\{\xi^k\}$ generated by the ADMM (i.e., by applying the map $\Gamma$ sequentially) converges to the unique optimal solution $\xi^\star$ (which is the fixed point of $\Gamma$). I suspect $\Gamma$ is a contraction in the sense that:
$$ \left\|\Gamma(\xi) - \xi^\star \right\| \leq \alpha \left\|\xi - \xi^\star \right\|$$
for any $\xi$, with $0 < \alpha < 1$.
However, I couldn't find any paper with such a result. I searched for papers on linear convergence of ADMM but couldn't find any relevant results (perhaps I didn't look carefully enough).
I would appreciate any pointer to such a result (either a direct result or relevant results that can be used to show it).
 A: Let's simplify a little and assume the constraint is $x-z=0$, that is $A=B=I$ and $c=0$. Also, let's assume that $x^k,z^k,u^k \in \mathbb{R}^n$ for all $k \in \mathbb{N}$. Then the ADMM iterates should be
\begin{align}
 x^{k+1} &= \arg\min_{x} \rho^{-1} f(x) + \frac12 \| x - z^k + u^k\|_2^2 = \text{prox}_{\rho^{-1} f}(z^k - u^k) \\
 z^{k+1} &= \arg\min_{z} \rho^{-1} g(z) + \frac12 \| z - x^{k+1} - u^{k}\|_2^2 = \text{prox}_{\rho^{-1} g}(x^{k+1} - u^k) \\
u^{k+1} &= u^k + x^{k+1} - z^{k+1}.
\end{align}
It follows by induction that the sequence $(x^k,z^k,u^k)_{k \in \mathbb{N}}$ can be written in terms of a single variable sequence $(w^k)_{k \in \mathbb{N}}$ defined by $w^{k+1}=\Phi(w^k)$, where
\begin{equation}
\Phi = \frac12 \text{Id} + \frac12 ( 2\text{prox}_{\rho^{-1}f} -\text{Id}) \circ  ( 2\text{prox}_{\rho^{-1}g} -\text{Id}). \tag{$\star$}
\end{equation}
In fact, it can be shown that
\begin{align}
z^k &= \text{prox}_{\rho^{-1} g} (w^k) = \alpha(w^k) \\
u^k &= (\text{Id} - \text{prox}_{\rho^{-1} g}) (w^k) = \beta(w^k) \\
x^{k+1} &= \text{prox}_{\rho^{-1} f} \circ ( 2 \text{prox}_{\rho^{-1} g} - \text{Id})(w^k) = \gamma(w^k)
\end{align}
for all $k \ge 1$. For a proof of this result, see Lemma $6.5$ of [1].
With this construction, using standard facts from convex analysis, we can show that $\Phi$ as defined in $(\star)$ is not necessarily a contraction, but actually a firmly-nonexpansive mapping, which is slightly weaker, but also very useful. To see that, you first remember that every proximal operator of a convex lower semicontinuous function is firmly-nonexpansive [2], and then we conclude that the mappings
\begin{align}
2\text{prox}_{\rho^{-1} f} - \text{id} &= (1-2) \text{id} + 2\text{prox}_{\rho^{-1} f} \\
2\text{prox}_{\rho^{-1} g} - \text{id} &= (1-2) \text{id} + 2\text{prox}_{\rho^{-1} g}
\end{align}
are nonexpansive (See Proposition $4.25 (ii)$ of [3]), and this means that the composition $(2\text{prox}_{\rho^{-1} f} - \text{id}) \circ (2\text{prox}_{\rho^{-1} g} - \text{id})$ is also nonexpansive. Finally, it follows that $\Phi$ is $1/2$-averaged, which is equivalent to firmly-nonexpansive.
Assuming that $\Phi$ has at least one fixed point (which can actually be proven), and using the Krasnoselskii-Mann Theorem, we have that the sequence $(w^k)_{k \in \mathbb{N}}$ converges to a fixed point of  $\Phi$. Let's say $w^* \leftarrow w^k$ is the fixed point we are looking for. Then, the point
\begin{align}
\left(\alpha(w^*),\beta(w^*),\gamma(w^*)\right)
\end{align}
should be a fixed point for $\Gamma$. The ADMM objective function convergence is also discussed in [1], in a slightly different setting. This reference is the only one I found which studies the ADMM convergence via fixed point iterations, and I would be very grateful if you or someone else would comment any other reference on this topic.
EDIT: I noticed that the writing of the ADMM iterations in terms of $(\star)$ follows easily from the equivalence of ADMM and Douglas-Rachford Splitting (DRS). See Section $9.1$ of [4].

*

*[1] Pravin Nair, Ruturaj G. Gavaskar, Kunal N. Chaudhury, Fixed-Point and Objective Convergence of Plug-and-Play Algorithms, 2021.

*[2] https://math.stackexchange.com/a/1900110/370982

*[3] Heinz H. Bauschke, Patrick L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces.

*[4] Ernest K. Ryu, Jialin Liu, Sicheng Wang, Xiaohan Chen, Zhangyang Wang, Wotao Yin, Plug-and-Play Methods Provably Converge with Properly Trained Denoisers.

