LemmaStatement: AI-adaptedProof: AI-generatedPipeline-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.

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The safeguarded PR+ direction has a uniform angle bound

Statement

Let $g \neq 0$ and let $d^{c a n d}$ be the PR+ candidate direction from Fletcher–Reeves and safeguarded Polak–Ribière nonlinear conjugate gradients. The safeguarded procedure either keeps $d^{c a n d}$ when $g^{T} d^{c a n d} \leq - σ ∥ g ∥_{2}^{2}, ∥ d^{c a n d} ∥_{2} \leq C ∥ g ∥_{2},$ or restarts to $d = - g$ , where $σ, C > 0$ . The resulting direction is strictly descending and its angle $θ$ with $- g$ satisfies $\cos θ \geq δ_{P R} : = \min {1, σ / C} > 0,$ using the angle convention of Wolfe and strong Wolfe line-search conditions.

Facts & Assumptions

Given: $g \neq 0$ , $σ, C > 0$ , the PR+ candidate, and the safeguard in the Statement.

[F1]

Fletcher–Reeves and safeguarded Polak–Ribière nonlinear conjugate gradients: the PR+ safeguard keeps a candidate only if $g^{T} d \leq - σ ∥ g ∥_{2}^{2}$ and $∥ d ∥_{2} \leq C ∥ g ∥_{2}$ ; otherwise it sets $d = - g$ .

[F2]

Wolfe and strong Wolfe line-search conditions: $\cos θ = - g^{T} d / (∥ g ∥_{2} ∥ d ∥_{2})$ for a descent direction.

[F3]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $∣ u^{T} v ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ for Euclidean vectors.

Proof

Given: The hypotheses and notation in the Statement.

1.1givenF1F2algebra

If the candidate is kept, then $g^{T} d \leq - σ ∥ g ∥_{2}^{2} < 0$ , so it is descending. Using the norm test and [F2], $\cos θ = \frac{- g^{T} d}{∥ g ∥_{2} ∥ d ∥_{2}} \geq \frac{σ}{C} .$

2.1givenF1F3step 1.1algebra∎

In the kept branch, [F3] also gives $\cos θ \leq 1$ , so necessarily $σ / C \leq 1$ and the bound in step 1.1 implies $\cos θ \geq δ_{P R}$ . If the candidate is reset, then $d = - g$ , whence $g^{T} d = - ∥ g ∥_{2}^{2} < 0$ and $\cos θ = 1 \geq δ_{P R}$ . Both branches prove the claim.

Depends on

Used by

Safeguarded PR+ converges in gradient norm under strong Wolfe Theorem

Dependency tree · two levels

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, second edition (standard reference, not scraped)
Jean Charles Gilbert and Jorge Nocedal, Global Convergence Properties of Conjugate Gradient Methods for Optimization, SIAM Journal on Optimization 2(1), 21–42 (1992), DOI 10.1137/0802003 (standard reference, not scraped)