AlgorithmAlgorithm: AI-adaptedProof: Not applicablePipeline-generatedjudge 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.

Certified inexact proximal-gradient and FISTA updates

Definition

Use the finite proximal-gradient or FISTA recurrences of Proximal-gradient, FISTA, and fixed-period restarted FISTA, but replace each exact proximal output $x_{k} = T_{L} (y_{k})$ by a reported finite point $x_{k} \in dom g$ and certificate $L (y_{k} - x_{k}) - \nabla f (y_{k}) \in \partial_{η_{k}} g (x_{k}), η_{k} \geq 0,$ where $\partial_{η} g$ is defined in Subdifferentials of proper extended-real convex functions. In proximal-gradient mode $y_{k} = x_{k - 1}$ ; in FISTA mode use the same momentum update as the exact recurrence. If the certificate cannot be supplied for a call, return status inconclusive and make no convergence claim for that run. The inexact rate theorems on this page use this certificate itself; they do not assert rates for an unverified objective-gap tolerance alone.

Depends on

Used by

Dependency tree · two levels

15 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, §3 and Appendix Lemma 2 (standard reference, not scraped)