(i) {α - Convex Optimization

Operator splitting schemes
Idea: replace explicit evaluation of the resolvent of ∂ (f 1 + f 2 ) ( i.e. J ∂(f1 +f 2 ))
,
by a sequence of calculations involving only ∂f 1 and ∂f 2 at a time.
An extensive literature essentially divided into three classes:
Splitting method Assumptions Iteration
Forward-Backward f 2 has a Lipschitzcontinuous
gradient
Backward-Backward f 1 ,f 2 proper lsc convex
but ...
Douglas/Peaceman-Rachford All f 1 ,f 2 proper lsc
convex
α t+1 = J µ∂f1 (I − µ∇f 2 )(α t )
α t+1 = J ∂f1 J ∂f2 (α t )
α t+1 =(J ∂f1 (2J ∂f2 − I)+I − J ∂f2 )(α t )
Stanford seminar 08-10