TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

A maximal-monotone resolvent is everywhere-defined and firmly nonexpansive

Statement

Let $A$ be maximal monotone on $E = R^{n}$ and let $ρ > 0$ . For every $y \in E$ there is a unique $x \in E$ with $y - x \in ρ A (x)$ . Thus the resolvent from Resolvent of a monotone relation is an everywhere-defined map and, for all $y, y^{'} \in E$ , $∥ J_{ρ A} y - J_{ρ A} y^{'} ∥_{2}^{2} \leq ⟨ J_{ρ A} y - J_{ρ A} y^{'}, y - y^{'} ⟩ .$ This is firm nonexpansiveness; it also implies the map is 1-Lipschitz.

Facts & Assumptions

Given: A maximal-monotone relation $A$ and $ρ > 0$ .

[A1]

The relation $A$ is maximal monotone and $ρ > 0$ .

[F1]

Monotone and maximal-monotone relations on Euclidean space: monotonicity means $⟨ x - x^{'}, a - a^{'} ⟩ \geq 0$ for graph pairs, and maximality excludes any proper monotone graph extension.

[F2]

Resolvent of a monotone relation: $x \in J_{ρ A} (y)$ iff $y = x + ρ a$ for some $a \in A (x)$ .

[F3]

One-point extension of a nonexpansive map in finite dimensions: a 1-Lipschitz map on any subset of $E$ admits a value at each added point satisfying all 1-Lipschitz inequalities.

[F4]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $∣ ⟨ v, w ⟩ ∣ \leq ∥ v ∥_{2} ∥ w ∥_{2}$ .

Proof

1.1A1F1constructalgebra

Transform every graph pair $(x, a) \in A$ into $p = x + ρ a$ and $q = x - ρ a$ . If two graph pairs are used, then $∥ q - q^{'} ∥_{2}^{2} \leq ∥ p - p^{'} ∥_{2}^{2},$ because the difference of the right and left sides is $4 ρ ⟨ x - x^{'}, a - a^{'} ⟩ \geq 0$ by [F1]. Also, if $p = p^{'}$ , this inequality forces $q = q^{'}$ . Thus the rule $T (p) = q$ is a well-defined 1-Lipschitz map on its range $D = {x + ρ a : (x, a) \in A}$ .

2.1A1F1F2F3step 1.1algebra

Fix arbitrary $y \in E$ . By [F3], extend $T$ at $y$ by some $q$ satisfying $∥ q - (x - ρ a) ∥_{2} \leq ∥ y - (x + ρ a) ∥_{2}$ for every $(x, a) \in A$ . Set $\tilde{x} = (y + q) / 2$ and $\tilde{a} = (y - q) / (2 ρ)$ . Expanding the squared inequality shows $⟨ \tilde{x} - x, \tilde{a} - a ⟩ \geq 0$ for every graph pair, so adjoining $(\tilde{x}, \tilde{a})$ preserves monotonicity. By maximality in [F1], it already belongs to $A$ . Hence $y = \tilde{x} + ρ \tilde{a}$ , proving existence. This also covers an initially empty graph: [F3] gives the extension and maximality rules out the empty graph.

3.1A1F1F2step 2.1algebra

If $x, x^{'}$ both satisfy $y - x \in ρ A (x)$ and $y - x^{'} \in ρ A (x^{'})$ , monotonicity gives $0 \leq ⟨ x - x^{'}, (y - x) - (y - x^{'}) ⟩ = - ∥ x - x^{'} ∥_{2}^{2}$ . Thus $x = x^{'}$ , so the resolvent is single-valued.

4.1A1F1F2F4step 3.1algebra∎

For inputs $y, y^{'}$ , write $x = J_{ρ A} y$ , $x^{'} = J_{ρ A} y^{'}$ . Monotonicity applied to $(x, (y - x) / ρ)$ and $(x^{'}, (y^{'} - x^{'}) / ρ)$ gives $∥ x - x^{'} ∥_{2}^{2} \leq ⟨ x - x^{'}, y - y^{'} ⟩ .$ Cauchy–Schwarz [F4] then gives $∥ x - x^{'} ∥_{2} \leq ∥ y - y^{'} ∥_{2}$ when $x \neq x^{'}$ , and the inequality is immediate when $x = x^{'}$ . This proves firm nonexpansiveness and the Lipschitz consequence.

Depends on

Used by

Dependency tree · two levels

21 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, Theorem 2 and §2 (standard reference, not scraped)
Neal Parikh and Stephen Boyd, Proximal Algorithms, §4.4 (standard reference, not scraped)