TheoremStatement: AI-adaptedProof: 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.

Critical directions are tangent to every positively weighted active inequality

Statement

Let $(x, ν, λ)$ satisfy the KKT conditions for the program of Inequality constrained programs, tangent cones, constraint qualifications, and KKT. For every $d \in L_{C} (x)$ ,

\nabla f (x)^{T} d = 0 ⟺ D g_{i} (x) d = 0 for every i \in I (x) with λ_{i} > 0.

Thus the critical cone is the linearized cone with equality imposed on each positively weighted active inequality.

Facts & Assumptions

Given: A KKT tuple and a vector in its linearized cone.

[F1]

KKT stationarity is $\nabla f + D h^{T} ν + D g^{T} λ = 0$ , with $λ \geq 0$ and $λ_{i} g_{i} (x) = 0$ (Inequality constrained programs, tangent cones, constraint qualifications, and KKT).

Proof

1.1givenF1algebra

For $d \in L_{C} (x)$ , the equality term in stationarity vanishes, and inactive multipliers vanish by complementarity. Therefore $\nabla f (x)^{T} d = - \sum_{i \in I (x)} λ_{i} D g_{i} (x) d$ . Every summand on the right is nonnegative because $λ_{i} \geq 0$ and $D g_{i} (x) d \leq 0$ .

2.1step 1.1algebra∎

The sum in step 1.1 is zero exactly when each term with $λ_{i} > 0$ is zero; because each such term is nonnegative, this is equivalent to $D g_{i} (x) d = 0$ for every positively weighted active index. That is the claimed equivalence and critical-cone description.

Depends on

Inequality constrained programs, tangent cones, constraint qualifications, and KKT

Used by

Dependency tree · two levels

4 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

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