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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 $y$?

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

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