The alternating direction method of multipliers (ADMM) algorithm solves problems of the form,
The iterates of ADMM are defined by the augmented Lagrangian,
and consist of,
Note that the method of multipliers algorithm considers the update of jointly, whereas ADMM takes a Gauss-Seidel pass. An often more convenient scaled form may be obtained by completing the square with the dual variable and residual (), and defining the scaled dual variable .
With the scaled dual variable, our augmented Lagrangian is written as,
and the ADMM iterates are expressed as,
Scaled Form ADMM
The ADMM iterates written in scaled form are often more amenable to being written with proximal operators.