LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.
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A Wolfe step gives positive Bfgs curvature on a descent direction

Statement

Let $f : D \to R$ be $C^{1}$ on an open $D \subseteq R^{n}$ . At $x \in D$ , let $p \neq 0$ be a strict descent direction, so $g^{T} p < 0$ for $g : = \nabla f (x)$ . Suppose $α > 0$ , $x + α p \in D$ , and $α$ satisfies the Wolfe curvature condition of Wolfe and strong Wolfe line-search conditions. With $s : = α p$ and $y : = \nabla f (x + α p) - \nabla f (x)$ , one has $s^{T} y \geq (1 - c_{2}) α ∣ g^{T} p ∣ > 0.$

Facts & Assumptions

Given: $f, x, p, α, g, s, y$ and the strict-descent and Wolfe hypotheses in the Statement.

[F1]

Wolfe and strong Wolfe line-search conditions: for $ϕ (t) = f (x + t p)$ , a Wolfe step satisfies $ϕ^{'} (α) \geq c_{2} ϕ^{'} (0)$ with $0 < c_{2} < 1$ ; the chain rule gives $ϕ^{'} (t) = \nabla f (x + t p)^{T} p$ .

[F2]

Bfgs curvature pairs and Hessian/inverse updates: $s = α p$ and $y = \nabla f (x + α p) - \nabla f (x)$ .

[F3]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $u^{T} v$ denotes the Euclidean inner product of $u, v \in R^{n}$ .

Proof

1.1givenF1F2F3algebra

By [F1] and [F2], $s^{T} y = α p^{T} (\nabla f (x + α p) - g) \geq α (c_{2} - 1) g^{T} p = (1 - c_{2}) α ∣ g^{T} p ∣ .$ The last quantity is strictly positive because $α > 0$ , $c_{2} < 1$ , and $g^{T} p < 0$ .

2.1step 1.1F2∎

Thus $s^{T} y > 0$ , so the curvature denominator used in the BFGS updates is positive and the pair meets the positive-curvature condition of Bfgs curvature pairs and Hessian/inverse updates.

Depends on

Used by

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, 2nd ed. (standard reference, not scraped)
Manuel Carreira-Perpiñán, Lecture Notes on Numerical Optimization (standard reference, not scraped)