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Douglas–Rachford splitting for two maximal-monotone relations

Definition

Given maximal-monotone $A, B$ on $E = R^{n}$ , $ρ > 0$ , and $w_{0} \in E$ , define for $k \geq 0$ $p_{k} = J_{ρ B} (w_{k}), q_{k} = J_{ρ A} (2 p_{k} - w_{k}), w_{k + 1} = w_{k} + q_{k} - p_{k} .$ The shadow is $p_{k}$ . Each exact update exists and is unique by A maximal-monotone resolvent is everywhere-defined and firmly nonexpansive. This recurrence specifies exact real arithmetic; a finite implementation that cannot evaluate a resolvent must report failure and must not advance the exact recurrence. No termination or convergence claim is part of this algorithm definition.

Facts & Assumptions

Given: Maximal-monotone $A, B$ , $ρ > 0$ , and a state $w_{k} \in E$ .

[A1]

Both relations are maximal monotone, $ρ > 0$ , and $w_{0} \in E$ .

[F1]

Resolvent of a monotone relation: $J_{ρ A} (y) = {x : y - x \in ρ A (x)}$ , and similarly for $B$ .

[F2]

A maximal-monotone resolvent is everywhere-defined and firmly nonexpansive: each such resolvent is a unique-valued map defined on all of $E$ .

Proof

1.1A1givenF1F2

Since $w_{k} \in E$ , [F2] gives a unique $p_{k}$ and then a unique $q_{k}$ at the input $2 p_{k} - w_{k}$ . The displayed formula therefore defines exactly one next state $w_{k + 1}$ .

2.1A1givenF1F2given∎

If a numerical routine does not return those exact resolvent values, the recurrence has not been completed; the item makes no convergence or approximate-solve assertion. The shadow sequence is precisely the first resolvent output $p_{k}$ .

Depends on

Used by

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Sources

Jonathan Eckstein and Dimitri P. Bertsekas, On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators, §4 (standard reference, not scraped)
Neal Parikh and Stephen Boyd, Proximal Algorithms, §4.4 (standard reference, not scraped)