I have a question regarding the theory behing solving Lasso's optimization problem via ADMM.

Let's look at this pdf https://candes.su.domains/teaching/math301/Lectures/Consensus.pdf.

First rewrite the problem (26.1) as (26.2), by adding the variable $$z$$, this is clear.

What is $$y$$ in the augmented Lagrangian (just one row below 26.2 in the pdf)? The expression is $$\lVert Ax-b\rVert^2 + \lambda |z| + \rho \langle y, z-x\rangle + \frac{\rho}2 \lVert z-x\rVert^2$$ (let's call it (3).

Specifically how do we go from the generic Langrangian formulation $$f(x) + g(z) + \rho |x-z| + \frac{\rho}2 \lVert z-x\rVert^2$$ to (3), which contains the multiplication with $$y$$?

• Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction.
– D.W.
Jan 4 at 23:59
• thank you, I've edited the question
– Rock
Jan 5 at 18:10