(i) {α - Convex Optimization
(i) {α - Convex Optimization
(i) {α - Convex Optimization
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Operator splitting schemes<br />
Idea: replace explicit evaluation of the resolvent of ∂ (f 1 + f 2 ) ( i.e. J ∂(f1 +f 2 ))<br />
,<br />
by a sequence of calculations involving only ∂f 1 and ∂f 2 at a time.<br />
An extensive literature essentially divided into three classes:<br />
Splitting method Assumptions Iteration<br />
Forward-Backward f 2 has a Lipschitzcontinuous<br />
gradient<br />
Backward-Backward f 1 ,f 2 proper lsc convex<br />
but ...<br />
Douglas/Peaceman-Rachford All f 1 ,f 2 proper lsc<br />
convex<br />
α t+1 = J µ∂f1 (I − µ∇f 2 )(α t )<br />
α t+1 = J ∂f1 J ∂f2 (α t )<br />
α t+1 =(J ∂f1 (2J ∂f2 − I)+I − J ∂f2 )(α t )<br />
J ∂fi<br />
= (I + ∂f i ) −1<br />
Moreau(-Yosida) proximity operators<br />
Stanford seminar 08-10