(i) {α - Convex Optimization

DR to solve Problem (P σ )
Theorem 1 Suppose that A = HΦ is a tight frame, i.e. AA ∗ = cI. Let µ ∈
(0, +∞), let {β t } be a sequence in (0, 2), and let {a t } and {b t } be sequences in H
such that ∑ t β t(2 − β t ) = +∞ and ∑ t β t (‖a t ‖ + ‖b t ‖) < +∞. Fix α 0 ∈ B l2 ,σ
and define the sequence of iterates,
α t+1/2 = α t + c −1 A ∗ (P Bσ − I) A(α t )+b
(
t
α t+1 = α t + β t
(prox µΨ ◦ 2α t+1/2 − α t) + a t − α t+1/2) ,
where,
(P Bσ − I)(x) =
σ
‖x−y‖ l2
As before
⎧
⎨0 if ‖x − y‖ l2
≤ σ,
(
)
⎩
σ
x + y 1 −
‖x−y‖
otherwise.
l2
Then {α t ,t≥ 0} converges weakly to some point α and α + c −1 A ∗ (P Bσ − I)(α)
is a solution to (P σ ).
Stanford seminar 08-19