(i) {α - Convex Optimization
(i) {α - Convex Optimization (i) {α - Convex Optimization
Characterization Theorem 1 (i) Existence: (P1) possesses at least one solution if f 1 + f 2 is coercive, i.e., lim ‖α‖→+∞ f 1 (α)+f 2 (α) = +∞. (ii) Uniqueness: (P1) possesses at most one solution if f 1 + f 2 is strictly convex. This occurs in particular when either f 1 or f 2 is strictly convex. (iii) Characterization: Let α ∈ H. Then the following statements are equivalent: (a) α solves (P1). (b) α =(I + ∂(f 1 + f 2 )) −1 (α) (proximal iteration). ∂f i is the subdifferential (set-valued map), a maximal monotone operator. J ∂(f1 +f 2 ) = (I + ∂(f 1 + f 2 )) −1 is the resolvent of ∂(f 1 + f 2 ) (firmly nonexpansive operator). Explicit inversion difficult in general Stanford seminar 08-9
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
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Characterization<br />
Theorem 1<br />
(i) Existence: (P1) possesses at least one solution if f 1 + f 2 is coercive, i.e.,<br />
lim ‖α‖→+∞ f 1 (α)+f 2 (α) = +∞.<br />
(ii) Uniqueness: (P1) possesses at most one solution if f 1 + f 2 is strictly convex.<br />
This occurs in particular when either f 1 or f 2 is strictly convex.<br />
(iii) Characterization: Let α ∈ H. Then the following statements are equivalent:<br />
(a) α solves (P1).<br />
(b) α =(I + ∂(f 1 + f 2 )) −1 (α) (proximal iteration).<br />
∂f i is the subdifferential (set-valued map), a maximal monotone operator.<br />
J ∂(f1 +f 2 )<br />
= (I + ∂(f 1 + f 2 )) −1 is the resolvent of ∂(f 1 + f 2 ) (firmly nonexpansive<br />
operator).<br />
Explicit inversion difficult in general<br />
Stanford seminar 08-9
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