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.

Truncated conjugate gradients with boundary and negative-curvature exits

Definition

Input a model of Quadratic trust-region models and the actual-to-predicted reduction ratio, a finite inner cap $J \in N_{0}$ , and a residual tolerance $τ_{t o l} \geq 0$ . If $g = 0$ , return $0$ with status stationary. Otherwise initialize $z_{0} = 0$ , $r_{0} = g$ , $d_{0} = - r_{0}$ , and $j = 0$ . For at most $J$ inner iterations:

If $∥ r_{j} ∥_{2} \leq τ_{t o l}$ , take $z_{j}$ as the candidate and stop.
If $d_{j} = 0$ , take $z_{j}$ as the candidate and stop; the final comparison below then returns $p_{C}$ whenever $z_{j}$ does not give at least its model decrease.
Put $c_{j} : = d_{j}^{T} B d_{j}$ . If $c_{j} \leq 0$ , set $t_{j} : = \frac{- ⟨ z_{j}, d_{j} ⟩ + \sqrt{⟨ z_{j}, d_{j} ⟩^{2} + ∥ d_{j} ∥_{2}^{2} (Δ^{2} - ∥ z_{j} ∥_{2}^{2})}}{∥ d_{j} ∥_{2}^{2}},$ take $z_{j} + t_{j} d_{j}$ as the candidate, and stop. This is the nonpositive curvature boundary exit.
Otherwise put $α_{j} : = ∥ r_{j} ∥_{2}^{2} / c_{j}$ and $\hat{z} : = z_{j} + α_{j} d_{j}$ . If $∥ \hat{z} ∥_{2} \geq Δ$ , use the same positive-root formula in step 3, take $z_{j} + t_{j} d_{j}$ as the candidate, and stop at the trust boundary.
Set $r_{j + 1} : = r_{j} + α_{j} B d_{j}$ and $z_{j + 1} : = \hat{z}$ . If $∥ r_{j + 1} ∥_{2} \leq τ_{t o l}$ , take $z_{j + 1}$ as the candidate and stop. Otherwise set $β_{j + 1} : = \frac{∥ r_{j + 1} ∥_{2}^{2}}{∥ r_{j} ∥_{2}^{2}}, d_{j + 1} : = - r_{j + 1} + β_{j + 1} d_{j},$ and increase $j$ by one.

If the cap is reached, take the current $z_{j}$ as the candidate. In all nonstationary cases compute the Cauchy step $p_{C}$ and compare model values: return the candidate if its model value is no greater than $m (p_{C})$ , and otherwise return $p_{C}$ . Thus a capped or unhelpful inner recurrence still returns the selected candidate or Cauchy step.

Depends on

Used by

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)
Nick Gould, Continuous Optimization Algorithms, Part 3.2: Trust-Region Methods (standard reference, not scraped)