Equivalent circuit for Linearized ADMM?

[antonysigma]

Hi, I have a question about linearization of (centralized) ADMM method. What is the equivalent circuit for the linearization step? Also called "cost function augmentation" depending on the discipline.

In particular, I wish to derive (aka reverse engineer) the equivalent circuits for the following paragraph:

Assuming the linear operator K \equiv I, I tried adding an RC circuit having resistance 1/rho and capacitance C. I think I am close, but still couldn't get the desired L-ADMM formulae.

$$\begin{align*} \frac{1}{C}\frac{de}{dt} &= \mu(x - e) - \lambda & \text{Node e}\\\ \mu(e - x) + \rho(z-x) &\in \partial g (x) & \text{Node x}\\\ \rho(x - z) + \lambda &\in \partial f (z) & \text{Node z}\\\ \frac{d\lambda}{dt} &=\frac{1}{L}(e - z) & \text{Inductor }L \end{align*}$$

What am I missing?