DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

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.

Scaled KKT residual and convex normal cone

Definition

Use the program and dimensions of Inequality constrained programs, tangent cones, constraint qualifications, and KKT. For any $x \in U$ , $ν \in R^{p}$ , and $λ \in R^{m}$ , define coordinatewise $[v]_{i}^{+} = \max {v_{i}, 0}$ and the scaled residual

r_{K K T} (x, ν, λ) = ∥ \nabla f (x) + D h (x)^{T} ν + D g (x)^{T} λ ∥_{2} + ∥ h (x) ∥_{2} + ∥ [g (x)]^{+} ∥_{2} + ∥ [- λ]^{+} ∥_{2} + ∥ λ \circ g (x) ∥_{2} .

The symbol $\circ$ denotes coordinatewise multiplication. The norm is the Euclidean norm from The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , and the norm of the zero-dimensional vector is $0$ .

For a convex set $C \subseteq R^{n}$ and $x \in C$ , its (outward) normal cone is

N_{C} (x) = {v \in R^{n} : ⟨ v, y - x ⟩ \leq 0 for every y \in C} .

This definition applies in particular when $C$ is closed and convex.

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

Boyd and Vandenberghe, Convex Optimization (standard reference, not scraped)
Nocedal and Wright, Numerical Optimization, second edition (standard reference, not scraped)