How to Minimize Augmented Lagrangian Function in ADMM for Lasso Problem - Solving ADMM Sub Problems The lasso problem for ADMM can be written as follow:
\begin{equation}
    \underset{x}{\min} 
    \frac{1}{2}\|{Ax-b}\|^2_2+\lambda\|{z}\|_1, \text{subject to } x-z=0
\end{equation}
The Augmented Lagrangian for this minimization is 
\begin{equation}
    F(x,z,u)=\Bigl(\tfrac{1}{2}\|{Ax-b}\|^2_2+\lambda\|{z}\|_1+u^T(x-z)+\frac{\rho}{2}\|x-z\|^2_2\Bigr)
\end{equation}
then iterations for $x,z,u$ is done with minimizing $F$ with respect to $x,z$. Could you show me how we do the minimization for $x,z$ and get the following results. (I think the gradient of the $F$ is equal to zero is applied but I could not see how to get the solutions)
\begin{equation}
\begin{split}
    &x^{k+1}=(A^TA+\rho I)^{-1}(A^Tb+\rho(z^k-u^k)) \\
    &z^{k+1}=S_{\lambda/\rho}(x^{k+1}+u^k) \\
    &u^{k+1}=u^k+x^{k+1}-z^{k+1}
\end{split}
\end{equation}
 A: ADMM steps (from https://web.stanford.edu/~boyd/papers/pdf/admm_distr_stats.pdf) can be 
\begin{align*}
    {{x}}   
    &\leftarrow  \arg\min_{{{x}} }  f \left( {{x}}  \right)  +  {{u}} ^T  \left( {{{x}} }  -  {{z}} \right)  +  \frac{\rho}{2}  \left\|  {{{x}} }  -  {{z}}  \right\|_2^2  \\
    &\equiv \arg\min_{{{x}} }  f \left( {{x}}  \right)  + \frac{\rho}{2} \left\| {{{x}} }  -  {{z}} + {u} \right\|_2^2   \\ 
{{z}}  
    &\leftarrow  \arg\min_{{{z}}}  g\left( {{z}} \right)  +    {{u}} ^T \left( {{{x}} }  -  {{z}} \right)  +  \frac{\rho}{2} \left\|  {{{y}} }  -  {{x}}  \right\|_2^2  \\
    &\equiv  \arg\min_{{{{z}}}}  g\left( {{z}} \right)  +    \frac{\rho}{2} \left\|  {{{x}} }  -  {{z}}  +  {u} \right\|_2^2  \\
    {{u}}  &\leftarrow {{u}}  +  \left( {{{x}} } - {{z}} \right) 
\end{align*}
Let us say $f(x) = \frac{1}{2} \|A x - b \|_2^2$ and $g(z) = \lambda \| z\|_1$. We can exploit proximal operator, that is,

Definition. Let $f: {\rm dom}_f \mapsto \left(-\infty\right., \left. +\infty \right]$ be a closed convex proper function, then 
  \begin{align*}
    {\rm prox}_{\lambda f}\left( x\right) := \left({I} + \lambda \partial f \right)^{-1} \left( x \right) = \arg\min_{u \in  {\rm dom}_f} \left\{ f\left({u}\right) + \frac{1}{ 2\lambda} \left\|x - u \right\|_2^2\right\} .
\end{align*}

Also, define for brevity (we can use the equivalent scaled-form ), 
$$F(x) := f \left( {{x}}  \right)  + \frac{\rho}{2} \left\| {{{x}} }  -  {{z}} + {u} \right\|_2^2$$ 
$$G(z) := g \left( {{z}}  \right)  + \frac{\rho}{2} \left\| {{{x}} }  -  {{z}} + {u} \right\|_2^2 .$$ 
Now, just find the gradients and set them to zero, that is, 
$$\frac{\partial F(x)}{\partial x} = 0 \Longleftrightarrow  \frac{1}{\rho}\partial f(x) + \left(x - z + u \right) = 0 \Longleftrightarrow x = \left(I + \frac{1}{\rho} \partial f \right)^{-1} \left( z - u\right) = \operatorname{prox}_{\frac{1}{\rho} f}\left( z - u\right)$$
and 
$$\frac{\partial G(z)}{\partial z} = 0 \Longleftrightarrow z = \left(I + \frac{1}{\rho} \partial g \right)^{-1} \left( x + u\right) = \operatorname{prox}_{\frac{1}{\rho} g}\left( x + u\right).$$
Thus, the ADMM iterative steps are
\begin{align*}
 {{x}^{k+1}} &:= \operatorname{prox}_{\frac{1}{\rho}f}\left( z^{k} - u^{k} \right)      \\ 
 {{z}^{k+1}}  &:= \operatorname{prox}_{\frac{1}{\rho}g}\left( {{x}^{k+1}} + u^{k} \right)    \\
 {{u}^{k+1}}  &:= {{u}^k}  +  \left( {{x}^{k+1}}  - {{z}^{k+1}} \right) 
\end{align*}
Now, you can use the prox operators for both affine $f(x)$ and L1 norm $g(z)$. 

Appendix
The prox operators for  $f(x) = \frac{1}{2} \|A x - b \|_2^2$ and $g(z) = \lambda \| z\|_1$ are given below.
\begin{align}
\operatorname{prox}_{\lambda f}\left( x \right) 
&= \arg\min_{v} \left\{ \frac{1}{2} \|A v - b \|_2^2 + \frac{1}{ 2 \lambda} \left\|x - v \right\|_2^2\right\} \\
\Longrightarrow 0&= A^T\left( Av - b \right) + \left(-\frac{1}{ \lambda} \left( x - v \right) \right) \\
\Longleftrightarrow 0&= \left(A^TA + \frac{1}{ \lambda}I \right)v - \left(A^Tb + \frac{1}{ \lambda} x \right)\\
\Longleftrightarrow v&= \operatorname{prox}_{\lambda f}\left( x \right) = \left(A^TA + \frac{1}{ \lambda} I \right)^{-1}\left(A^Tb + \frac{1}{ \lambda} x \right).
\end{align}
\begin{align}
\operatorname{prox}_{\lambda g}\left( z \right) 
&= \arg\min_{v} \left\{ \lambda \| v\|_1 + \frac{1}{ 2} \left\|z - v \right\|_2^2\right\} \\
&=  \arg\min_{ \left\{v_i\right\}} \left\{  \sum_i \lambda|v_i| + \frac{1}{ 2} \sum_i \left\|z_i - v_i \right\|_2^2\right\}
\end{align}
Since the problem is separable, then you can use KKT conditions to obtain so-called soft thresholding operator. Not to make this post too long, I can refer to you for instance this The Proximal Operator of the $ {L}_{1} $ Norm Function which shows the derivation. 
I hope this helps you. 
