DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Subgradients and the subdifferential of a convex function

Definition

Let $f : C \to R$ be convex on a convex set $C \subseteq R^{n}$ (Convex and strictly convex functions on Euclidean convex sets), and let $a \in C$ . A vector $v$ is a subgradient of $f$ at $a$ when $f (y) \geq f (a) + ⟨ v, y - a ⟩$ for every $y$ in the domain.

The subdifferential is the set

$\partial f (a) : = {v \in R^{n} : f (y) \geq f (a) + ⟨ v, y - a ⟩ for every y \in C},$

with the Euclidean inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . The definition permits $\partial f (a)$ to be empty or to contain more than one vector.

Depends on

Used by

Dependency tree · two levels

17 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

D. Bertsekas, MIT 6.253 Convex Analysis and Optimization, Lecture 12 (standard reference, not scraped)
D. Drusvyatskiy, Convex Analysis and Nonsmooth Optimization, §3.5 (standard reference, not scraped)