LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Douglas–Rachford fixed-point map is firmly nonexpansive

Statement

For maximal-monotone $A, B$ and $ρ > 0$ , let $T = I - J_{ρ B} + J_{ρ A} (2 J_{ρ B} - I) .$ Then $T : E \to E$ is firmly nonexpansive: $∥ T (w) - T (w^{'}) ∥_{2}^{2} \leq ⟨ T (w) - T (w^{'}), w - w^{'} ⟩ .$

Facts & Assumptions

Given: Maximal-monotone $A, B$ , $ρ > 0$ , and $w, w^{'} \in E$ .

[A1]

Both relations are maximal monotone, $ρ > 0$ , and the inputs lie in $E$ .

[F1]

Douglas–Rachford splitting for two maximal-monotone relations: the update map is $T (w) = w + J_{ρ A} (2 J_{ρ B} w - w) - J_{ρ B} w$ .

[F2]

A maximal-monotone resolvent is everywhere-defined and firmly nonexpansive: each resolvent $J$ obeys $∥ J v - J v^{'} ∥^{2} \leq ⟨ J v - J v^{'}, v - v^{'} ⟩$ .

Proof

1.1A1F2algebra

For any firmly nonexpansive $J$ , its reflector $R = 2 J - I$ is nonexpansive: with $d = v - v^{'}$ and $e = J v - J v^{'}$ , [F2] gives $∥ 2 e - d ∥_{2}^{2} = ∥ d ∥_{2}^{2} + 4 ∥ e ∥_{2}^{2} - 4 ⟨ e, d ⟩ \leq ∥ d ∥_{2}^{2} .$ Thus $R_{A} R_{B}$ is nonexpansive because it is a composition of two nonexpansive maps.

2.1A1F1step 1.1algebra∎

Expanding the reflectors gives $T = (I + R_{A} R_{B}) / 2$ . For $d = w - w^{'}$ and $s = R_{A} R_{B} w - R_{A} R_{B} w^{'}$ , nonexpansiveness gives $∥ s ∥_{2} \leq ∥ d ∥_{2}$ , and ${∥ \frac{d + s}{2} ∥}_{2}^{2} \leq ⟨ \frac{d + s}{2}, d ⟩$ is equivalent after multiplying by $4$ and cancelling common terms to $∥ s ∥_{2}^{2} \leq ∥ d ∥_{2}^{2}$ . This is the firm inequality for $T$ .

Depends on

Used by

Douglas–Rachford shadows converge to a zero of the operator sum Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)