Augmented Lagranian Solver

 min (y,x)  1/2 ||f(y)||_2^2 + lambda Phi(x) s.t. y=x

using an augmented lagrangian approach with alternating direction

the lagrangian for this subproblem is of the form

L(y,x,Multipler,lambda) = 1/2 ||f(y)||_2^2 + lambda Phi(x) + tau/2 ||y-x-Multiplier/tau||_2^2

L2 subproblem:

fun is a function handle for the L2 subproblem and has the form function y=fun(tau,b,eps) and should solve the L2 subproblem

min y 1/2 ||f(y)||_2^2 + tau/2 || y - b ||_2^2 with accuracy eps

L1 subproblem:

the solution to the L1 subproblem depends on the regularizer, Phi(x), but is of the form

min x Phi(x) + tau/2 || x - b ||_2^2