The following comes from my own derivations, but the results can be see in  equation (7).
Suppose we have a vector and want to project it onto to intersection of two sets . This projection may not have a closed form, but if projection onto each set separately is cheap/closed-form, we're in luck. We begin by converting our starting formulation to something ADMM-able via indicator functions,
Our scaled-form augmented Lagrangian is then,
with scaled dual variable . From the Lagrangian we can see that our -update is straightforward, but the -update requires some manipulation. If we complete the square w.r.t ,
we see that its update is also a projection, with inputs scaled by the step-size (note is only a constant w.r.t ).
The resulting ADMM algorithm for the 2-set best approximation problem is as follows, with , ,
This is reminiscent of Dykstra's projection algorithm and POCS, but different from Dykstra's we have a step-size to play with for accelerated convergence.
| ||The proximal operator of indicator function of a set is orthogonal-projection onto . |