AlgorithmAlgorithm: 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.

Full-step exact-Hessian SQP after active-set identification

Definition

Let $c_{A} (x) = (h (x), g_{I} (x))$ stack all equalities and the identified active inequalities, and let $μ = (ν, λ_{I})$ . For a current pair $(x, μ)$ , form $F_{A} (x, μ) = (\binom{\nabla_{x} L (x, ν, λ_{I}, 0_{I^{c}})}{c_{A} (x)}), J_{A} (x, μ) = [\begin{array}{cc} \nabla_{x x}^{2} L (x, ν, λ_{I}, 0_{I^{c}}) & D c_{A} (x)^{T} \\ D c_{A} (x) & 0 \end{array}] .$ If $J_{A}$ is nonsingular, solve $J_{A} (\binom{p}{δ μ}) = - F_{A} (x, μ)$ and return the full Newton pair $(x + p, μ + δ μ)$ . A failed rank test or solve returns kkt-solve-failure. The active set is supplied to this step; this algorithm has no line search and makes no global convergence claim.

Facts & Assumptions

Given: The program convention of Inequality constrained programs, tangent cones, constraint qualifications, and KKT, an identified active set $I$ , and a current primal-dual pair.

[F1]

The linearized quadratic subproblem in sequential quadratic programming defines the affine constraint model and the quadratic-model KKT block convention; the exact Lagrangian Hessian used here is specified in this algorithm.

[F2]

The SQP KKT matrix is nonsingular under positive reduced curvature gives the nonsingularity criterion for the displayed saddle-point matrix.

[F3]

Inequality constrained programs, tangent cones, constraint qualifications, and KKT fixes the active-set Lagrangian $L = f + ν^{T} h + λ^{T} g$ and the nonnegative inequality-multiplier convention.

[A1]

The active set is identified, the active rows are independent, and the exact Lagrangian Hessian is positive on their null space whenever a step is claimed.

Proof

1.1givenF1F3algebra

Differentiating $F_{A}$ with respect to $(x, μ)$ gives the block matrix $J_{A}$ : the derivative of stationarity in $x$ is the exact Lagrangian Hessian, its derivative in $μ$ is $D c_{A} (x)^{T}$ , and the derivative of the constraint block is $D c_{A} (x)$ . Thus the linear solve is precisely the Newton equation for $F_{A} = 0$ .

2.1givenA1F2step 1.1algebra∎

When the active rows are independent and the Lagrangian Hessian is positive on their null space, [F2] makes that Newton equation uniquely solvable. Otherwise the algorithm reports failure and does not return a Newton step; a returned pair is the stated full step.

Depends on

Used by

Dependency tree · two levels

6 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

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, 2nd ed. (standard reference, not scraped)
Ralf Herzog, Algorithms and Preconditioning in PDE-Constrained Optimization, lecture notes (standard reference, not scraped)