LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Optimality relations for exact two-block ADMM subproblems

Statement

For any exact two-block ADMM sweep from Scaled two-block ADMM with explicit capped-solve statuses, put $Δ z^{k + 1} = z^{k + 1} - z^{k}$ and $y^{k + 1} = ρ u^{k + 1}$ . Then $- K^{*} (y^{k + 1} - ρ L Δ z^{k + 1}) \in \partial f (x^{k + 1}), - L^{*} y^{k + 1} \in \partial g (z^{k + 1}) .$

Facts & Assumptions

Given: Proper convex $f, g$ , linear $K, L$ , $ρ > 0$ , and an exact attained sweep.

[A1]

The algorithm's proper closed convex objectives and linear maps are in force, and both subproblem minimizers are attained exactly.

[F1]

Scaled two-block ADMM with explicit capped-solve statuses: the exact $x$ and $z$ subproblems and update $y^{k + 1} = y^{k} + ρ r^{k + 1}$ are as displayed there.

[F2]

Subdifferentials of proper extended-real convex functions: for a proper convex extended-real function and a finite-valued point $x$ , $a \in \partial f (x)$ iff $f (v) \geq f (x) + ⟨ a, v - x ⟩$ for every $v$ ; the same convention applies to $g$ .

[F3]

Every linear map between finite-dimensional inner product spaces has a unique adjoint gives the unique adjoints of the finite-dimensional Euclidean maps $K, L$ ; The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ : $⟨ K x, y ⟩ = ⟨ x, K^{*} y ⟩$ and similarly for $L$ .

Proof

1.1A1F1F2F3givenalgebra

Write the $x$ -subproblem objective as $f (x) + Q (x)$ , where $\nabla Q (x^{k + 1}) = ρ K^{*} (K x^{k + 1} + L z^{k} - c + u^{k})$ by [F3]. For any $v \in dom f$ and $0 < t < 1$ , convexity bounds $f (x^{k + 1} + t (v - x^{k + 1}))$ above by $(1 - t) f (x^{k + 1}) + t f (v)$ . Minimality and differentiability of $Q$ , after dividing the resulting nonnegative objective difference by $t$ and letting $t ↓ 0$ , give $f (v) \geq f (x^{k + 1}) - ⟨ \nabla Q (x^{k + 1}), v - x^{k + 1} ⟩$ . For $v \notin dom f$ this inequality is automatic, so $- \nabla Q (x^{k + 1}) \in \partial f (x^{k + 1})$ by [F2].

2.1A1F1F2F3step 1.1algebra∎

Since $K x^{k + 1} + L z^{k} - c = r^{k + 1} - L Δ z^{k + 1}$ and $y^{k + 1} = ρ (u^{k} + r^{k + 1})$ , the gradient in step 1.1 equals $K^{*} (y^{k + 1} - ρ L Δ z^{k + 1})$ . This proves the first inclusion. Applying the same segment argument to the $z$ -subproblem gives $- ρ L^{*} (K x^{k + 1} + L z^{k + 1} - c + u^{k}) \in \partial g (z^{k + 1})$ ; the bracket times $ρ$ is $y^{k + 1}$ , proving the second inclusion.

Depends on

Used by

Dependency tree · two levels

14 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

Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein, Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers, §3.1 (standard reference, not scraped)
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)