LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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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.

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The SQP KKT matrix is nonsingular under positive reduced curvature

Statement

Let $A \in R^{r \times n}$ have full row rank, where $0 \leq r \leq n$ , and let $H \in R^{n \times n}$ be symmetric and positive definite on $\ker A$ : $z^{T} H z > 0$ whenever $0 \neq z \in \ker A$ . Then $K = [\begin{array}{cc} H & A^{T} \\ A & 0 \end{array}]$ is nonsingular. Here the zero-row case $r = 0$ is allowed.

Facts & Assumptions

Given: The matrix hypotheses in the Statement and the SQP KKT block convention of The linearized quadratic subproblem in sequential quadratic programming.

[A1]

$A$ has full row rank, and $H$ is positive definite on $\ker A$ , as stated.

[F1]

The linearized quadratic subproblem in sequential quadratic programming stacks equality rows before active inequality rows in the displayed saddle matrix. Independence is the full-row-rank hypothesis [A1], not a consequence of that definition.

Proof

1.1givenF1algebra

Suppose $K (\binom{z}{μ}) = 0$ . Its lower block gives $A z = 0$ , so $z \in \ker A$ ; multiplying the upper block $H z + A^{T} μ = 0$ by $z^{T}$ gives $z^{T} H z = 0$ . Positive definiteness on $\ker A$ forces $z = 0$ .

2.1givenstep 1.1algebra∎

With $z = 0$ , the upper block gives $A^{T} μ = 0$ . Full row rank of $A$ means $A^{T}$ is injective, so $μ = 0$ . Thus the square matrix $K$ has trivial kernel and is nonsingular; when $r = 0$ , the same argument says $H$ has trivial kernel.

Depends on

The linearized quadratic subproblem in sequential quadratic programming

Used by

Dependency tree · two levels

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, 2nd ed. (standard reference, not scraped)