(i) {α - Convex Optimization
(i) {α - Convex Optimization (i) {α - Convex Optimization
Characterizing Problem (Peq) min Ψ(α) s.t. y = HΦα. Proposition 1 (i) Existence: (P eq ) has at least one solution. (ii) Uniqueness: (P eq ) has a unique solution if ψ is strictly convex. (iii) If α solves (P eq ), then it solves (P σ=0 ). Stanford seminar 08-20
DR to solve Problem (Peq) Theorem 1 Let A = HΦ. 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 and define the sequence of iterates, α t+1/2 = α t + A ∗ (AA ∗ ) −1 ( y − Aα t) + b t ( α t+1 = α t + β t (prox µΨ ◦ 2α t+1/2 − α t) + a t − α t+1/2) . Then {α t , t ≥ 0} converges weakly to some point α and α+A ∗ (AA ∗ ) −1 (y − Aα) is a solution to (P eq ). As before Stanford seminar 08-21
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DR to solve Problem (Peq)<br />
Theorem 1 Let A = HΦ. Let µ ∈ (0, +∞), let {β t } be a sequence in (0, 2),<br />
and let {a t } and {b t } be sequences in H such that ∑ t β t(2 − β t ) = +∞ and<br />
∑<br />
t β t (‖a t ‖ + ‖b t ‖) < +∞. Fix α 0 and define the sequence of iterates,<br />
α t+1/2 = α t + A ∗ (AA ∗ ) −1 ( y − Aα t) + b t<br />
(<br />
α t+1 = α t + β t<br />
(prox µΨ ◦ 2α t+1/2 − α t) + a t − α t+1/2) .<br />
Then {α t , t ≥ 0} converges weakly to some point α and α+A ∗ (AA ∗ ) −1 (y − Aα)<br />
is a solution to (P eq ).<br />
As before<br />
Stanford seminar 08-21
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