LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

Uniform Hessian bounds control gradient changes and Taylor remainders

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, and let $f : U \to R$ have continuous first and second partial derivatives. Put $g (x) : = \nabla f (x)$ and $G (x) : = \nabla^{2} f (x)$ , using the conventions of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and The Hessian matrix and critical points of a scalar field. Let $K \subseteq U$ be convex. Suppose that for every $x \in K$ , $G (x)$ is symmetric and, for constants $0 < μ \leq Λ < \infty$ , $μ ∥ w ∥_{2}^{2} \leq w^{T} G (x) w (w \in R^{n}), ∥ G (x) ∥_{2} \leq Λ,$ where $∥ A ∥_{2} : = \sup_{∥ w ∥_{2} = 1} ∥ A w ∥_{2}$ . Then for all $u, v \in K$ , with $d : = u - v$ , $∥ g (u) - g (v) ∥_{2} \leq Λ ∥ d ∥_{2}, (g (u) - g (v))^{T} d \geq μ ∥ d ∥_{2}^{2},$ and $\frac{μ}{2} ∥ d ∥_{2}^{2} \leq f (u) - f (v) - g (v)^{T} d \leq \frac{Λ}{2} ∥ d ∥_{2}^{2} .$ Separately, let $W \subseteq U$ be convex, let $x_{*} \in W$ , and suppose $G$ is $L_{G}$ -Lipschitz on $W$ in operator norm. Then for all $u, v \in W$ , with $d : = u - v$ , $∥ g (u) - g (v) - G (x_{*}) d ∥_{2} \leq L_{G} \max {∥ u - x_{*} ∥_{2}, ∥ v - x_{*} ∥_{2}} ∥ d ∥_{2} .$

Facts & Assumptions

Given: The open domain, convex set, continuous partial-derivative assumptions, Hessian bounds and notation in the Statement.

[F1]

The Hessian matrix and critical points of a scalar field and The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: $G (x)$ is the Hessian matrix of second partial derivatives and $g (x)$ is the gradient vector of first partial derivatives.

[F2]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $w^{T} z$ is the Euclidean inner product and $∣ w^{T} z ∣ \leq ∥ w ∥_{2} ∥ z ∥_{2}$ .

[F3]

If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative: continuous partial derivatives make a map totally differentiable with derivative matrix equal to its Jacobian.

[F4]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : the derivative of a composition is the composition of its total derivatives.

[F5]

If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz: if a vector-valued function is continuous on $[a, b]$ , differentiable on $(a, b)$ , and its derivative norm is at most $M$ , then its endpoint difference has norm at most $M (b - a)$ .

[F6]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : for a real function continuous on $[a, b]$ and differentiable on $(a, b)$ , some $c \in (a, b)$ satisfies $h (b) - h (a) = h^{'} (c) (b - a)$ .

[F7]

Clairaut--Schwarz theorem for continuous second partial derivatives: because $f$ is $C^{2}$ on the open set $U$ , its mixed second partial derivatives commute. Under [F1], this identifies the Jacobian of $g = \nabla f$ with the Hessian matrix $G$ at every point of $U$ .

Proof

Given: The hypotheses and notation in the Statement. Fix $u, v \in K$ , put $d : = u - v$ , and define $z (t) : = v + t d$ for $0 \leq t \leq 1$ .

1.1givenF1F3F4F7construct

Convexity gives $z (t) \in K$ . For each point of the compact segment, openness gives a neighborhood on which the path remains in $U$ ; a finite subcover supplies an open interval containing $[0, 1]$ on which the composition is defined. By [F3], $g$ is differentiable along the segment with derivative matrix its Jacobian; [F7] identifies that Jacobian with $G$ . The chain rule [F4] then gives $(g \circ z)^{'} (t) = G (z (t)) d$ , while $ϕ (t) : = f (z (t))$ satisfies $ϕ^{'} (t) = g (z (t))^{T} d$ and $ϕ^{''} (t) = d^{T} G (z (t)) d$ .

2.1givenF5step 1.1algebra

The operator-norm bound gives $∥ G (z (t)) d ∥_{2} \leq Λ ∥ d ∥_{2}$ . Apply [F5] to the vector path $t \mapsto g (z (t))$ on $[0, 1]$ to get $∥ g (u) - g (v) ∥_{2} \leq Λ ∥ d ∥_{2}$ .

2.2givenF6step 1.1algebra

Put $q (t) : = g (z (t))^{T} d$ . By step 1.1, $q^{'} (t) = d^{T} G (z (t)) d \geq μ ∥ d ∥_{2}^{2}$ . Applying [F6] to $q$ on $[0, 1]$ gives $q (1) - q (0) = q^{'} (c)$ for some $c \in (0, 1)$ , and hence $(g (u) - g (v))^{T} d \geq μ ∥ d ∥_{2}^{2}$ .

2.3givenF6step 1.1algebrachoose

Define $ψ (t) : = ϕ (t) - ϕ (0) - t ϕ^{'} (0) - \frac{μ}{2} t^{2} ∥ d ∥_{2}^{2}$ . Then $ψ (0) = ψ^{'} (0) = 0$ and $ψ^{''} (t) = d^{T} G (z (t)) d - μ ∥ d ∥_{2}^{2} \geq 0$ . For each $t \in (0, 1]$ , [F6] applied to $ψ^{'}$ on $[0, t]$ gives $ψ^{'} (t) - ψ^{'} (0) = t ψ^{''} (c_{t}) \geq 0$ ; continuity handles $t = 0$ . A second application of [F6], now to $ψ$ on $[0, 1]$ , gives $ψ (1) - ψ (0) = ψ^{'} (c) \geq 0$ . Thus $f (u) - f (v) - g (v)^{T} d \geq \frac{μ}{2} ∥ d ∥_{2}^{2}$ .

3.1givenF2F6step 1.1step 2.3algebrachoose

Since $∣ d^{T} G (z (t)) d ∣ \leq ∥ d ∥_{2} ∥ G (z (t)) d ∥_{2} \leq Λ ∥ d ∥_{2}^{2}$ by [F2], define $χ (t) : = ϕ (0) + t ϕ^{'} (0) + \frac{Λ}{2} t^{2} ∥ d ∥_{2}^{2} - ϕ (t)$ . Then $χ (0) = χ^{'} (0) = 0$ and $χ^{''} (t) = Λ ∥ d ∥_{2}^{2} - d^{T} G (z (t)) d \geq 0$ . Applying [F6] first to $χ^{'}$ on $[0, t]$ and then to $χ$ on $[0, 1]$ as in step 2.3 gives $χ (1) \geq 0$ , proving the upper Taylor bound.

4.1givenF3F4F5F7algebra∎

Under the additional hypotheses on $W$ , the path $q (t) : = g (v + t d) - g (v) - t G (x_{*}) d$ has derivative $q^{'} (t) = (G (v + t d) - G (x_{*})) d$ by the chain rule and the Jacobian identity [F7]. Convexity and the triangle inequality give $∥ v + t d - x_{*} ∥_{2} \leq (1 - t) ∥ v - x_{*} ∥_{2} + t ∥ u - x_{*} ∥_{2} \leq \max {∥ u - x_{*} ∥_{2}, ∥ v - x_{*} ∥_{2}}$ . Therefore $∥ q^{'} (t) ∥_{2} \leq L_{G} \max {∥ u - x_{*} ∥_{2}, ∥ v - x_{*} ∥_{2}} ∥ d ∥_{2}$ . Applying [F5] to $q$ proves the last estimate in the Statement.

Depends on

Used by

Dependency tree · two levels

57 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)
Manuel Carreira-Perpiñán, Lecture Notes on Numerical Optimization (standard reference, not scraped)