AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Dogleg steps for positive-definite trust-region models

Definition

Input a quadratic trust-region model of Quadratic trust-region models and the actual-to-predicted reduction ratio with positive-definite $B$ (Positive definite, negative definite, semidefinite, and indefinite quadratic forms) and radius $Δ > 0$ . If $g = 0$ , return $p = 0$ with status stationary. Otherwise compute the Cauchy step $p_{C}$ by The Cauchy step for a quadratic trust-region model and the Newton-model step $p_{N}$ by solving $B p_{N} = - g$ .

If $∥ p_{N} ∥_{2} \leq Δ$ , return $p_{N}$ .
If $∥ p_{C} ∥_{2} = Δ$ , return $p_{C}$ .
Otherwise put $d : = p_{N} - p_{C}$ , $a : = ∥ d ∥_{2}^{2}$ , $b : = 2 ⟨ p_{C}, d ⟩$ , and $c : = ∥ p_{C} ∥_{2}^{2} - Δ^{2} < 0$ . Compute $t : = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} .$ Set $p_{D} : = p_{C} + t d$ , the boundary intersection on the segment from the Cauchy step to the Newton-model step.
Compare the two model values. Return $p_{D}$ if $m (p_{D}) \leq m (p_{C})$ and return $p_{C}$ otherwise.

The status is step in the nonstationary branches.

Depends on

Used by

Positive-definite Newton and dogleg steps are well-defined Lemma

Dependency tree · two levels

8 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, second edition (standard reference, not scraped)