LemmaStatement: 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.

One-step estimate for a certified inexact proximal call

Statement

Let $f$ be convex, continuously differentiable and have $L$ -Lipschitz gradient, $L > 0$ , and let $g$ be proper, closed and convex. Given $y$ , suppose $x^{+}$ and $η \geq 0$ satisfy $L (y - x^{+}) - \nabla f (y) \in \partial_{η} g (x^{+}) .$ With $F = f + g$ , for every $z \in R^{n}$ , $F (z) \geq F (x^{+}) + \frac{L}{2} ∥ x^{+} - y ∥_{2}^{2} + L ⟨ y - z, x^{+} - y ⟩ - η .$

Facts & Assumptions

Given: The functions, certificate and points in the Statement.

[F1]

Certified inexact proximal-gradient and FISTA updates: each reported proximal call supplies $L (y - x^{+}) - \nabla f (y) \in \partial_{η} g (x^{+})$ with $η \geq 0$ .

[F2]

Subdifferentials of proper extended-real convex functions: the certificate means $g (z) - g (x^{+}) \geq ⟨ L (y - x^{+}) - \nabla f (y), z - x^{+} ⟩ - η$ for finite $g (z)$ .

[F3]

Descent inequality for a Lipschitz gradient: $f (x^{+}) \leq f (y) + ⟨ \nabla f (y), x^{+} - y ⟩ + (L / 2) ∥ x^{+} - y ∥_{2}^{2}$ .

[F4]

Differentiable convex functions are characterized by the gradient inequality: $f (z) \geq f (y) + ⟨ \nabla f (y), z - y ⟩$ .

[F5]

A finite continuous convex summand preserves proper closed convexity: $F = f + g$ is a well-defined proper convex extended-real objective.

[F6]

Subdifferentials of proper extended-real convex functions: membership in $\partial_{η} g (x^{+})$ entails $x^{+} \in dom g$ ; since $f$ is real-valued, $F (x^{+})$ is finite.

Proof

1.1F1F2F3F4F5F6givenalgebra

Put $d = x^{+} - y$ . If $g (z) = + \infty$ , then $F (z) = + \infty$ and the asserted inequality holds because [F6] makes its right side finite. Otherwise $F (z)$ and $F (x^{+})$ are real, so ordinary subtraction is valid. From [F3] and [F4], $F (z) - F (x^{+}) \geq g (z) - g (x^{+}) + ⟨ \nabla f (y), z - x^{+} ⟩ - \frac{L}{2} ∥ d ∥_{2}^{2} .$ By [F2], the first two terms on the right are bounded below by $L ⟨ y - x^{+}, z - x^{+} ⟩ - η$ .

2.1step 1.1F6algebra∎

Since $y - x^{+} = - d$ and $z - x^{+} = (z - y) - d$ , the lower bound from step 1.1 is $L ⟨ y - x^{+}, z - x^{+} ⟩ - \frac{L}{2} ∥ d ∥_{2}^{2} - η = L ⟨ y - z, d ⟩ + \frac{L}{2} ∥ d ∥_{2}^{2} - η .$ Adding the finite value $F (x^{+})$ gives the displayed inequality.

Depends on

Used by

Dependency tree · two levels

25 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

Mark Schmidt, Nicolas Le Roux, and Francis Bach, Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization, Appendix §§6.1–6.2 (standard reference, not scraped)
Neal Parikh and Stephen Boyd, Proximal Algorithms, §4.2 (standard reference, not scraped)