I am solving the Lasso problem

minimize 1/2*|| Ax - b ||_2^2 + lambda || x ||_1

with CVX and getting good results for my study but when I apply ADMM algorithm I am getting very different results. Why is there such a difference? I would like to understand the solution of the problem better but cant follow what CVX does.

Answers and suggestions will be very helpful.

Thank you in advance.