I'm trying to express the following constrained optimization problem:
\begin{equation} \begin{aligned} & \underset{x}{\text{minimize}} & & \| b - x \|_1 + \lambda \| \mathbf{D}x \|_2 \\ & \text{subject to} & & x \leq b \\ \end{aligned} \end{equation}
Where $\mathbf{D}$ is a finite-difference matrix and $b$ is some observed data.
I want to use ADMM, which handles only equality constraint. From reading round, I learned that you can convert it into an inequality constraint by imposing a non-negativity constraint on $-x + b$:
$ 0 \leq -x + b $
I've re-formulated the problem to now become:
\begin{equation} \begin{aligned} & \underset{x}{\text{minimize}} & & \| z_1 \|_1 + \lambda \| z_2 \|_2 + I_{C}(z_3)\\ & \text{subject to} & -x + b &= z_1 \\ & & \mathbf{D}x &= z_2\\ & & -x + b &= z_3 \end{aligned} \end{equation}
Where the $I_C(z_3)$ is an indicator function to impose non-negative constraint on $z_3$. But somehow this looks 'wrong' because I don't know if it's legitimate to define the same auxilliary variable twice in the constraint.... What is the correct way to re-formulate the original problem so that I can use ADMM?