(i) {α - Convex Optimization

FB to solve Problem (P τ )
Theorem 1 Let {µ t ,t ∈ N} be a sequence such that 0 < inf t µ t ≤ sup t µ t <
2/ ‖HΦ‖ 2 , let {β t ,t∈ N} be a sequence in (0, 1], and let {a t ,t∈ N} and {b t ,t∈ N}
be sequences in H such that ∑ t ‖a t‖ < +∞ and ∑ t ‖b t‖ < +∞. Fix α 0 , and
define the sequence of iterates:
α t+1 = α t + β t
(
proj BΨ,τ
(
α t + µ t
(
Φ ∗ H ∗ ( y − HΦα t)) − b t
)
+ at − α (t))
where proj BΨ,τ
is the projector onto the Ψ-ball of radius τ.
(i) {α (t) ,t≥ 0} converges weakly to a minimizer of (P λ ).
(ii) For Ψ(α) =‖α‖ l1
, there exists a subsequence {α (t) ,t ≥ 0} that converges
strongly to a minimizer of (P λ ).
(iii) The projection operator proj BΨ,τ
is
⎧
⎨α if Ψ (α) ≤ τ,
proj BΨ,τ
(α) =
⎩prox κΨ α otherwise.
,
where κ (depending on α and τ) is chosen such that Ψ (prox κΨ (α)) = τ.
Stanford seminar 08-16