I have attempted to implement the coordinate descent algorithm for a separably convex problem of the form
$$\min \sum f_i(x_i) \\ \text{s.t.} \ Ax = b $$
using the augmented Lagrangian $$L(x,\lambda) = \sum f_i(x_i) + \lambda'(Ax-b) + \rho\|Ax-b\|^2$$
Expressing the norm term as $\|Ax-b\|_2^2 = \sum_i \Big[\tilde{A}_{ii}x_i^2 + x_i\sum_{j \neq i}\tilde{A}_{ij}x_j \Big] -2\sum_i x_i\sum_j A_{ij}b_j +b'b $, where $\tilde{A} := A'A$, we can rewrite the Lagrangian as
$$L(x,\lambda) = -\lambda'b + \rho b'b + \sum_{i=1}^n \Big( f_i(x_i) + \rho \tilde{A}_{ii}x_i^2 + x_i\Big[ \lambda' A_i - 2\rho b' A_i + \rho \sum_{j \neq i} \tilde{A}_{ij} x_j \Big ]\Big)$$
Now, as I understand it, we should be able to cyclically minimize $L_i(x_i,\lambda)$ while treating $x_j$ for $j \neq i$ as constant. That is the gist of coordinate descent, right? But I'm just not getting the right results. I've confirmed that I haven't made any mistake in rewriting the $\|Ax-b\|$ term. The minimization solver seems to be correct as well.
This makes me suspect I've made some conceptual error. What is my misunderstanding here?